Algebra 2 Quiz: Synthetic Division

20 questions · exam conditions

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Synthetic DivisionQuestion 1 of 20

To compare methods: long division works for any polynomial divisor, but synthetic division is faster when the divisor is $(x - c)$ . Divide $P(x)=x^3+x^2-5x+2$ by $(x-1)$ using synthetic division (so $c=1$ ). Read the quotient coefficients and remainder from the bottom row (remainder $=P(1)$ ).

Quotient

x^2+0x-5

, remainder

2

Quotient

x^2+2x-3

, remainder

0

Quotient

x^2+2x-4

, remainder

1

Quotient

x^2+2x-3

, remainder

-1

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Algebra 2 Quiz

Algebra 2 Quiz: Synthetic Division

Practice Synthetic Division in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Synthetic Division, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Quotient $x^2+0x-5$ , remainder $2$
Quotient $x^2+2x-3$ , remainder $0$
Quotient $x^2+2x-4$ , remainder $1$
Quotient $x^2+2x-3$ , remainder $-1$ (correct answer)

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form

(x - c)

that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is

(x - c)

: instead of the complex long division setup, you just write the value c (from

x - c

) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. The process is: (1) bring down the first coefficient, (2) multiply it by c and write under the next coefficient, (3) add that column, (4) repeat multiply-and-add until done. To divide

x^3 + x^2 - 5x + 2

(x - 1)

using synthetic division: (1) Identify c = 1. (2) Write coefficients: 1, 1, -5, 2. (3) Bring down 1, multiply by 1 for 1, add to 1 for 2; multiply 2 by 1 for 2, add to -5 for -3; multiply -3 by 1 for -3, add to 2 for -1. (4) Bottom row: 1, 2, -3, -1, so quotient

x^2 + 2x - 3

with remainder

-1

, and

P(1) = 1 + 1 - 5 + 2 = -1

confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder from the bottom row accurately. Choice C incorrectly assumes remainder 0, perhaps from misadding the final column—always double-check the last addition since it gives the remainder directly. Synthetic division step-by-step: (1) From divisor

(x - c)

, identify c (solve

x - c = 0

to get

x = c

). (2) Write coefficients of polynomial in descending degree order, using 0 for any missing degrees. (3) Draw shape: c on left outside, coefficients across top. (4) Algorithm: bring down first coefficient to bottom row, multiply by c and write result under next coefficient, add column to get next bottom row value, repeat until done. (5) Read: bottom row has quotient coefficients (one less degree than original) with last value as remainder—practice makes this fast! Why synthetic division is worth learning: it's 3-5 times faster than long division for

(x - c)

divisors, produces the same answer, and makes testing zeros via the Remainder Theorem super efficient.

Question 2

Use synthetic division (more efficient than long division for $(x-c)$ ) to divide $P(x)=x^3-2x^2-5x+6$ by $(x-3)$ . Use $c=3$ and coefficients $1,-2,-5,6$ . Read the quotient from the bottom row and the remainder from the last entry (equal to $P(3)$ ). Which is correct?

Quotient $x^2-x-2$ , remainder $0$
Quotient $x^2+x-2$ , remainder $6$
Quotient $x^2+x-2$ , remainder $0$ (correct answer)
Quotient $x^2+x+2$ , remainder $0$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division efficiently processes the coefficients through a multiply-and-add pattern, yielding both quotient and remainder in one calculation. To divide

x^3 - 2x^2 - 5x + 6

(x - 3)

: (1) c = 3, coefficients: 1, -2, -5, 6. (2) Bring down 1. (3) 1×3 = 3, add to -2: -2+3 = 1. (4) 1×3 = 3, add to -5: -5+3 = -2. (5) (-2)×3 = -6, add to 6: 6+(-6) = 0. Bottom row: 1, 1, -2, 0, giving quotient

x^2 + x - 2

with remainder 0. Choice A correctly identifies quotient

x^2 + x - 2

and remainder 0, verified by

P(3) = 27 - 18 - 15 + 6 = 0

. The zero remainder reveals that (x - 3) is a factor, so

x^3 - 2x^2 - 5x + 6 = (x - 3)(x^2 + x - 2)

, and we can factor further:

(x - 3)(x + 2)(x - 1)

! Synthetic division's efficiency shines when factoring: quickly test potential zeros, and when you find one (remainder = 0), you've reduced the problem to factoring a lower-degree polynomial. Combined with the Rational Zeros Theorem, this makes complete factorization systematic and achievable!

Question 3

Use synthetic division to test whether $(x+2)=(x-(-2))$ is a factor of $P(x)=x^4-2x^3-7x^2+8x+12.$ Use $c=-2$ with coefficients $1,-2,-7,8,12$ . Which statement is correct? (If the remainder is $0$ , then it is a factor.)

$(x+2)$ is not a factor; remainder $-24$
$(x+2)$ is a factor; remainder $0$ (correct answer)
$(x+2)$ is not a factor; remainder $24$
$(x+2)$ is a factor; remainder $-2$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To test if (x + 2) = (x - (-2)) is a factor of x⁴ - 2x³ - 7x² + 8x + 12 using synthetic division: (1) Identify c = -2 from divisor (x + 2). (2) Write coefficients: 1, -2, -7, 8, 12. (3) Setup: -2 on left, coefficients on top row. (4) Bring down 1. (5) Multiply 1 × (-2) = -2, write under -2, add: -2 + (-2) = -4. (6) Multiply -4 × (-2) = 8, write under -7, add: -7 + 8 = 1. (7) Multiply 1 × (-2) = -2, write under 8, add: 8 + (-2) = 6. (8) Multiply 6 × (-2) = -12, write under 12, add: 12 + (-12) = 0. (9) Bottom row: 1, -4, 1, 6, 0. The remainder is 0, so (x + 2) IS a factor! The Remainder Theorem confirms: P(-2) should equal 0, and checking: 16 + 16 - 28 - 16 + 12 = 0, yes! Choice B correctly identifies that (x + 2) is a factor with remainder 0. Choice A incorrectly claims it's not a factor with remainder -24—this would result from calculation errors or using wrong value of c. When remainder is 0, we've found a factor: x⁴ - 2x³ - 7x² + 8x + 12 = (x + 2)(x³ - 4x² + x + 6). This illustrates how synthetic division helps factor polynomials: test potential zeros (from Rational Zeros Theorem), and when you find one, you get both confirmation and the reduced polynomial for further factoring!

Question 4

Use synthetic division (instead of long division) to find the remainder when $P(x)=x^4-5x^2+3$ is divided by $(x+3)=(x-(-3)).$ Use $c=-3$ and remember to include a $0$ coefficient for the missing $x^3$ and $x$ terms. By the Remainder Theorem, the remainder equals $P(-3)$ . What is the remainder?

$33$
$-33$
$-39$
$39$ (correct answer)

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x⁴ - 5x² + 3 by (x + 3) = (x - (-3)) using synthetic division: (1) Identify c = -3 from divisor. (2) Write coefficients including zeros for missing terms: 1, 0, -5, 0, 3. (3) Setup: -3 on left, coefficients on top. (4) Bring down 1. (5) Multiply 1 × (-3) = -3, write under 0, add: 0 + (-3) = -3. (6) Multiply -3 × (-3) = 9, write under -5, add: -5 + 9 = 4. (7) Multiply 4 × (-3) = -12, write under 0, add: 0 + (-12) = -12. (8) Multiply -12 × (-3) = 36, write under 3, add: 3 + 36 = 39. Bottom row: 1, -3, 4, -12, 39. The remainder is 39. The Remainder Theorem confirms: P(-3) = 81 - 45 + 3 = 39, yes! Choice D correctly identifies the remainder as 39. Choice C's error of -39 suggests a sign mistake—synthetic division requires careful attention to signs when c is negative! Synthetic division step-by-step: remember to include 0 coefficients for any missing degree terms, and be extra careful with signs when c is negative. Why synthetic division is worth learning: it's 3-5 times faster than long division for (x - c) divisors and makes evaluating polynomials at specific values super efficient via the Remainder Theorem!

Question 5

Use synthetic division to test whether $(x+1)=(x-(-1))$ is a factor of $P(x)=x^3+2x^2-x-2.$ Use $c=-1$ with coefficients $1,2,-1,-2$ . The remainder (last entry) equals $P(-1)$ by the Remainder Theorem. Which statement is correct?

Remainder = 4, so $(x+1)$ is not a factor.
Remainder = 1, so $(x+1)$ is a factor.
Remainder = 0, so $(x+1)$ is a factor. (correct answer)
Remainder = -2, so $(x+1)$ is a factor.

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form

(x - c)

that's much faster than polynomial long division. To test if

(x + 1) = (x - (-1))

is a factor, use synthetic division with

c = -1

and check if the remainder equals 0. To divide

x^3 + 2x^2 - x - 2

(x + 1)

: (1)

c = -1

, coefficients: 1, 2, -1, -2. (2) Bring down 1. (3)

1 \times (-1) = -1

, add to 2:

2 + (-1) = 1

. (4)

1 \times (-1) = -1

, add to -1:

-1 + (-1) = -2

. (5)

(-2) \times (-1) = 2

, add to -2:

-2 + 2 = 0

. Bottom row: 1, 1, -2, 0, so remainder = 0. Choice A correctly states remainder = 0, therefore

(x + 1)

is a factor, which we verify:

P(-1) = -1 + 2 + 1 - 2 = 0

. The zero remainder means we can factor:

x^3 + 2x^2 - x - 2 = (x + 1)(x^2 + x - 2)

, and the quotient

x^2 + x - 2

factors further as

(x + 2)(x - 1)

! Synthetic division with

c = -1

requires extra care with signs—each multiplication by -1 changes the sign. When testing factors of form

(x + a)

, remember to use

c = -a

in synthetic division, then a zero remainder confirms it's a factor!

Question 6

Use synthetic division (more efficient than long division for $(x-c)$ ) to divide $P(x)=x^3-2x^2-5x+6$ by $(x-3).$ Use $c=3$ and coefficients $1,-2,-5,6$ . Which result is correct? (Remainder Theorem check: remainder $=P(3)$ .)

Quotient $x^2+x-2$ , remainder $0$ (correct answer)
Quotient $x^2+x-2$ , remainder $6$
Quotient $x^2-5x+10$ , remainder $0$
Quotient $x^2+x+2$ , remainder $0$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x³ - 2x² - 5x + 6 by (x - 3) using synthetic division: (1) Identify c = 3 from divisor (x - 3). (2) Write coefficients: 1, -2, -5, 6. (3) Setup: 3 on left, coefficients on top row. (4) Bring down 1. (5) Multiply 1 × 3 = 3, write under -2, add: -2 + 3 = 1. (6) Multiply 1 × 3 = 3, write under -5, add: -5 + 3 = -2. (7) Multiply -2 × 3 = -6, write under 6, add: 6 + (-6) = 0. (8) Bottom row: 1, 1, -2, 0. This means quotient is x² + x - 2 (degree 2) with remainder 0. So (x³ - 2x² - 5x + 6) divided by (x - 3) = x² + x - 2 exactly! The Remainder Theorem confirms: P(3) should equal 0, and checking: 27 - 18 - 15 + 6 = 0, yes! Choice A correctly executes the synthetic division algorithm and identifies quotient x² + x - 2 with remainder 0. Choice B incorrectly claims remainder is 6—if you stop the process early or misread the final value, you might think the remainder is the original constant term, but synthetic division transforms all values! Synthetic division reveals that (x - 3) is a factor since remainder is 0, so we can write x³ - 2x² - 5x + 6 = (x - 3)(x² + x - 2). Going further, we could factor x² + x - 2 = (x + 2)(x - 1), giving complete factorization: x³ - 2x² - 5x + 6 = (x - 3)(x + 2)(x - 1). This shows how synthetic division connects to finding all zeros of a polynomial!

Question 7

Use synthetic division to divide $P(x)=2x^4-3x^3+x^2-4x+6$ by $(x-1).$ Synthetic division is faster than long division here because the divisor is linear of the form $(x-c)$ . What are the quotient and remainder? (Remainder $=P(1)$ .)

Quotient $2x^3-x^2+0x-4$ , remainder $6$
Quotient $2x^3-x^2+0x-4$ , remainder $2$ (correct answer)
Quotient $2x^3-x^2+2x-2$ , remainder $4$
Quotient $2x^3-5x^2+6x-10$ , remainder $8$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide 2x⁴ - 3x³ + x² - 4x + 6 by (x - 1) using synthetic division: (1) Identify c = 1 from divisor (x - 1). (2) Write coefficients: 2, -3, 1, -4, 6. (3) Setup: 1 on left, coefficients on top. (4) Bring down 2. (5) Multiply 2 × 1 = 2, write under -3, add: -3 + 2 = -1. (6) Multiply -1 × 1 = -1, write under 1, add: 1 + (-1) = 0. (7) Multiply 0 × 1 = 0, write under -4, add: -4 + 0 = -4. (8) Multiply -4 × 1 = -4, write under 6, add: 6 + (-4) = 2. Bottom row: 2, -1, 0, -4, 2. This means quotient is 2x³ - x² + 0x - 4 (or 2x³ - x² - 4) with remainder 2. The Remainder Theorem confirms: P(1) = 2 - 3 + 1 - 4 + 6 = 2, yes! Choice B correctly identifies the quotient as 2x³ - x² + 0x - 4 with remainder 2. Choice A's error of remainder 6 would give P(1) = 6, but we calculated P(1) = 2! Synthetic division step-by-step: when a coefficient in the quotient is 0 (like the x term here), you can write it as 0x or omit it, but be careful about the degree. Why synthetic division is worth learning: it works perfectly even when some quotient coefficients are zero!

Question 8

Use synthetic division to divide $P(x)=2x^3-7x^2+4x-3$ by $(x-3).$ Set up with $c=3$ and coefficients $2,-7,4,-3$ . The bottom row gives the quotient coefficients and the final number is the remainder (also $P(3)$ by the Remainder Theorem). Which result is correct?

Quotient $2x^2-x+1$ , remainder $0$ (correct answer)
Quotient $2x^2-x+1$ , remainder $-2$
Quotient $2x^2+x+1$ , remainder $0$
Quotient $2x^2-x-1$ , remainder $0$

2x^3 - 7x^2 + 4x - 3

divided by

(x - 3)

, use c = 3 and coefficients 2, -7, 4, -3, then apply the multiply-and-add algorithm. To divide: (1) Bring down 2. (2)

2 \times 3 = 6

, add to -7:

-7+6 = -1

. (3)

(-1) \times 3 = -3

, add to 4:

4 + (-3) = 1

. (4)

1 \times 3 = 3

, add to -3:

-3+3 = 0

. Bottom row: 2, -1, 1, 0, giving quotient

2x^2 - x + 1

with remainder 0. Choice A correctly identifies the quotient as

2x^2 - x + 1

and remainder as 0, which we can verify:

P(3) = 54 - 63 + 12 - 3 = 0

. The zero remainder tells us something special:

(x - 3)

is a factor of

P(x)

, so we can write

2x^3 - 7x^2 + 4x - 3 = (x - 3)(2x^2 - x + 1)

. Synthetic division not only divides but also reveals factors—when remainder equals 0, you've found a factor! This makes synthetic division essential for polynomial factoring: test potential zeros using the Rational Zeros Theorem, and when you find one (remainder = 0), you've factored out one linear term and can continue factoring the quotient!

Question 9

Use synthetic division (instead of long division) to divide $P(x)=2x^4-x^3+3x^2-5x+6$ by $(x-1)$ (so $c=1$ ). Read the quotient coefficients and remainder from the bottom row, and remember the remainder equals $P(1)$ by the Remainder Theorem.

Quotient $2x^3-2x^2+x-4$ , remainder $10$
Quotient $2x^3+0x^2+4x-1$ , remainder $5$
Quotient $2x^3+x^2+4x-1$ , remainder $0$
Quotient $2x^3+x^2+4x-1$ , remainder $5$ (correct answer)

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form

(x - c)

that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is

(x - c)

: instead of the complex long division setup, you just write the value

c

(from

x - c

c

and write under the next coefficient, (3) add that column, (4) repeat multiply-and-add until done. To divide

2x^4 - x^3 + 3x^2 - 5x + 6

(x - 1)

using synthetic division: (1) Identify

c = 1

. (2) Write coefficients: 2, -1, 3, -5, 6. (3) Bring down 2, multiply by 1 for 2, add to -1 for 1; multiply 1 by 1 for 1, add to 3 for 4; multiply 4 by 1 for 4, add to -5 for -1; multiply -1 by 1 for -1, add to 6 for 5. (4) Bottom row: 2, 1, 4, -1, 5, so quotient

2x^3 + x^2 + 4x - 1

with remainder 5, and

P(1) = 2 - 1 + 3 - 5 + 6 = 5

confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder from the bottom row accurately. Choice B assumes remainder 0, possibly from a miscalculation in the final step—verify by plugging in

c

to the original polynomial. Synthetic division step-by-step: (1) From divisor

(x - c)

, identify

c

(solve

x - c = 0

to get

x = c

). (2) Write coefficients of polynomial in descending degree order, using 0 for any missing degrees. (3) Draw shape:

c

on left outside, coefficients across top. (4) Algorithm: bring down first coefficient to bottom row, multiply by

c

and write result under next coefficient, add column to get next bottom row value, repeat until done. (5) Read: bottom row has quotient coefficients (one less degree than original) with last value as remainder—practice makes this fast! Why synthetic division is worth learning: it's 3-5 times faster than long division for

(x - c)

divisors, produces the same answer, and makes testing zeros via the Remainder Theorem super efficient.

Question 10

Use synthetic division to divide $P(x)=x^4-2x^3+0x^2+5x-4$ by $(x+1)=(x-(-1)).$ Include the $0$ coefficient for the missing $x^2$ term. Read the quotient and remainder from the bottom row (remainder $=P(-1)$ ). Which result is correct?

Quotient $x^3-3x^2+3x+2$ , remainder $-6$ (correct answer)
Quotient $x^3-x^2-x+6$ , remainder $2$
Quotient $x^3-3x^2+3x+2$ , remainder $-8$
Quotient $x^3-x^2+0x+5$ , remainder $-4$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x⁴ - 2x³ + 0x² + 5x - 4 by (x + 1) = (x - (-1)) using synthetic division: (1) Identify c = -1 from divisor. (2) Write coefficients including the 0: 1, -2, 0, 5, -4. (3) Setup: -1 on left, coefficients on top. (4) Bring down 1. (5) Multiply 1 × (-1) = -1, write under -2, add: -2 + (-1) = -3. (6) Multiply -3 × (-1) = 3, write under 0, add: 0 + 3 = 3. (7) Multiply 3 × (-1) = -3, write under 5, add: 5 + (-3) = 2. (8) Multiply 2 × (-1) = -2, write under -4, add: -4 + (-2) = -6. Bottom row: 1, -3, 3, 2, -6. This means quotient is x³ - 3x² + 3x + 2 with remainder -6. The Remainder Theorem confirms: P(-1) = 1 + 2 + 0 - 5 - 4 = -6, yes! Choice A correctly identifies the quotient as x³ - 3x² + 3x + 2 with remainder -6. Choice C's error of remainder -8 suggests an arithmetic mistake in the final step—always double-check additions when working with negative numbers! Synthetic division step-by-step: don't forget to include 0 coefficients for missing terms, and be extra careful with signs when c is negative. Why synthetic division is worth learning: even with missing terms and negative divisors, it's still much faster and less error-prone than long division!

Question 11

Use synthetic division to divide $P(x)=x^4-2x^3+0x^2+5x-6$ by $(x+1)=(x-(-1)).$ Set up with $c=-1$ and coefficients $1,-2,0,5,-6$ (include the $0$ ). Read the quotient from the bottom row and the remainder as the last number (which equals $P(-1)$ ). Which result is correct?

Quotient $x^3-3x^2+3x+2$ , remainder $0$
Quotient $x^3-3x^2-3x+2$ , remainder $-8$
Quotient $x^3-x^2-x+6$ , remainder $-8$
Quotient $x^3-3x^2+3x+2$ , remainder $-8$ (correct answer)

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form

(x - c)

that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is

(x - c)

: instead of the complex long division setup, you just write the value c (from

(x - c)

) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide

x^4 - 2x^3 + 0x^2 + 5x - 6

(x + 1)

using synthetic division: identify

c = -1

, write coefficients

1, -2, 0, 5, -6

; bring down

1

, multiply by

-1

for

-1

, add to

-2

for

-3

; multiply

-3

-1

for

3

, add to

0

for

3

; multiply

3

-1

for

-3

, add to

5

for

2

; multiply

2

-1

for

-2

, add to

-6

for

-8

; quotient

x^3 - 3x^2 + 3x + 2

with remainder

-8

, and

P(-1) = 1 + 2 + 0 - 5 - 6 = -8

confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder accurately. Choice B assumes remainder

0

incorrectly: check the final addition carefully when using negative c. Synthetic division step-by-step: from divisor

(x - c)

, identify c; write coefficients in descending order, including zeros if needed; bring down first, multiply by c, add to next, repeat; read quotient from bottom row except last, which is remainder. You're mastering this—include those zero coefficients to avoid mistakes!

Question 12

Synthetic division is a streamlined method (compared to long division) for dividing by $(x-c)$ . Use synthetic division to divide $P(x)=x^3+x^2-5x+2$ by $(x-1).$ (Use $c=1$ with coefficients $1,1,-5,2$ .) The bottom row gives the quotient coefficients and the final entry is the remainder, which equals $P(1)$ by the Remainder Theorem. What is the quotient and remainder?

$Q(x)=x^2+2x-3,\ R=0$
$Q(x)=x^2-2x-3,\ R=1$
$Q(x)=x^2+2x-3,\ R=-1$ (correct answer)
$Q(x)=x^2+2x+3,\ R=-7$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x cubed + x squared -5x +2 by (x - 1): c=1, coefficients 1,1,-5,2. Bring down 1. 11=1, 1+1=2. 21=2, -5+2=-3. -3*1=-3, 2+ -3=-1. Bottom: 1,2,-3,-1. Q=x^2 +2x -3, R=-1. P(1)=1+1-5+2=-1, yes. Matches choice A. Choice B uses the wrong sign or arithmetic: it has Q=x^2 -2x -3, R=1, perhaps misadded columns. Synthetic division step-by-step: (1) Identify c. (2) List coefficients with zeros for missing terms. (3) Bring down, multiply by c, add, repeat. (4) Quotient from bottom except last, last is R. Keep practicing to avoid sign errors! This method is great for quick factoring and root testing.

Question 13

Use synthetic division to find the remainder when $P(x)=x^4-5x^2+3$ is divided by $(x+3)=(x-(-3)).$ Use $c=-3$ and include $0$ coefficients for missing terms (so coefficients are $1,0,-5,0,3$ ). What is the remainder? (Remainder Theorem: remainder $=P(-3)$ .)

$-39$
$39$ (correct answer)
$-120$
$120$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x⁴ - 5x² + 3 by (x + 3) = (x - (-3)) using synthetic division: (1) Identify c = -3 from divisor (x + 3). (2) Write coefficients including zeros for missing terms: 1, 0, -5, 0, 3. (3) Setup: -3 on left, coefficients on top row. (4) Bring down 1. (5) Multiply 1 × (-3) = -3, write under 0, add: 0 + (-3) = -3. (6) Multiply -3 × (-3) = 9, write under -5, add: -5 + 9 = 4. (7) Multiply 4 × (-3) = -12, write under 0, add: 0 + (-12) = -12. (8) Multiply -12 × (-3) = 36, write under 3, add: 3 + 36 = 39. (9) Bottom row: 1, -3, 4, -12, 39. The remainder is 39. The Remainder Theorem confirms: P(-3) should equal 39, and checking: 81 - 45 + 3 = 39, yes! Choice B correctly identifies the remainder as 39 from the synthetic division process. Choice A incorrectly claims remainder is -39, which would be a sign error—perhaps confusing the sign of c or making an arithmetic mistake during the multiply-and-add steps. Synthetic division requires careful attention to signs, especially when c is negative! When dividing by (x + 3), use c = -3, not c = 3. This is a common source of errors: from (x - c), we need the value that makes x - c = 0, so from (x + 3) = (x - (-3)), we get c = -3. Practice makes this automatic: (x - 5) gives c = 5, (x + 7) gives c = -7, (x - (-4)) gives c = -4!

Question 14

To compare methods, you could use long division, but synthetic division is more efficient for a divisor of the form $(x-c)$ . Use synthetic division to divide $x^3+x^2-5x+2$ by $(x-1).$ What is the quotient and remainder? (Remainder Theorem check: remainder $=P(1)$ .)

Quotient $x^2+2x-3$ , remainder $-1$ (correct answer)
Quotient $x^2+0x-5$ , remainder $2$
Quotient $x^2+2x-3$ , remainder $1$
Quotient $x^2+2x+3$ , remainder $-1$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x³ + x² - 5x + 2 by (x - 1) using synthetic division: (1) Identify c = 1 from divisor (x - 1). (2) Write coefficients: 1, 1, -5, 2. (3) Setup: 1 on left, coefficients on top row. (4) Bring down 1. (5) Multiply 1 × 1 = 1, write under 1, add: 1 + 1 = 2. (6) Multiply 2 × 1 = 2, write under -5, add: -5 + 2 = -3. (7) Multiply -3 × 1 = -3, write under 2, add: 2 + (-3) = -1. (8) Bottom row: 1, 2, -3, -1. This means quotient is x² + 2x - 3 (degree 2) with remainder -1. So (x³ + x² - 5x + 2) divided by (x - 1) = x² + 2x - 3 + (-1)/(x - 1). The Remainder Theorem confirms: P(1) should equal -1, and checking: 1 + 1 - 5 + 2 = -1, yes! Choice A correctly executes the synthetic division algorithm and reads the quotient x² + 2x - 3 and remainder -1 from the bottom row. Choice C has the wrong sign on the remainder—getting -1 instead of 1 requires careful attention to signs during the multiply-and-add steps. Synthetic division is indeed more efficient than long division for divisors of form (x - c): while long division requires writing out x terms and aligning like terms, synthetic division works purely with coefficients in a compact format. For a cubic divided by linear, synthetic division takes about 30 seconds while long division might take 2-3 minutes!

Question 15

A polynomial $f(x)$ has degree 4, and when divided by $(x + 3)$ using synthetic division, the bottom row of the synthetic division table shows: $1, -2, 5, -1, 4$ . Which statement about $f(x)$ is correct?

The leading coefficient of $f(x)$ is 4, and $f(-3) = 1$
The leading coefficient of $f(x)$ is 1, and $f(-3) = 4$
The quotient is $x^3 - 2x^2 + 5x - 1$ with remainder 4 (correct answer)
The quotient is $x^4 - 2x^3 + 5x^2 - x + 4$ with no remainder

Explanation: In synthetic division, the bottom row shows coefficients of the quotient followed by the remainder. Since f(x) has degree 4 and we're dividing by a linear factor, the quotient has degree 3. The first four numbers (1, -2, 5, -1) are coefficients of the quotient x³ - 2x² + 5x - 1, and the last number (4) is the remainder, which equals f(-3). Choice A confuses remainder with leading coefficient. Choice B reverses the leading coefficient and remainder. Choice D incorrectly makes the quotient degree 4.

Question 16

When using synthetic division to divide $P(x) = 2x^3 - 7x^2 + 8x - 3$ by $(x - 2)$ , a student obtains the quotient $Q(x) = 2x^2 - 3x + 2$ and remainder $R = 1$ . If $P(2) = k$ , what is the value of $k$ ?

$k = 0$ , indicating that $(x - 2)$ is a factor of $P(x)$
$k = 1$ , which matches the remainder from the division (correct answer)
$k = 6$ , obtained by evaluating $Q(2) + R$
$k = -1$ , since the signs alternate in synthetic division

Explanation: By the division algorithm, P(x) = (x - 2) · Q(x) + R. When x = 2, this becomes P(2) = (2 - 2) · Q(2) + R = 0 · Q(2) + R = R = 1. This is the Remainder Theorem: when dividing P(x) by (x - c), the remainder equals P(c). Choice A is wrong because P(2) ≠ 0. Choice C incorrectly adds Q(2) to the remainder. Choice D has no mathematical basis.

Question 17

Use synthetic division to divide $P(x)=x^3+x^2-5x+2$ by $(x-1).$ Because the divisor is $(x-c)$ , synthetic division is faster than long division. Set up with $c=1$ and coefficients $1,1,-5,2$ . What are the quotient and remainder (remainder equals $P(1)$ )?

Quotient $x^2+2x-3$ , remainder $0$
Quotient $x^2+2x-3$ , remainder $-1$ (correct answer)
Quotient $x^2-2x-3$ , remainder $-1$
Quotient $x^2+2x+3$ , remainder $-1$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide x^3 + x^2 - 5x + 2 by (x - 1) using synthetic division: identify c = 1, write coefficients 1, 1, -5, 2; bring down 1, multiply by 1 for 1, add to 1 for 2; multiply 2 by 1 for 2, add to -5 for -3; multiply -3 by 1 for -3, add to 2 for -1; quotient x^2 + 2x - 3 with remainder -1, and P(1) = 1 + 1 - 5 + 2 = -1 confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder accurately. Choice B incorrectly assumes remainder 0: verify with the Remainder Theorem to catch such errors. Synthetic division step-by-step: from divisor (x - c), identify c; write coefficients in descending order, including zeros if needed; bring down first, multiply by c, add to next, repeat; read quotient from bottom row except last, which is remainder. Keep up the excellent work—this method will save you so much time on tests!

Question 18

A student says synthetic division is more efficient than long division when the divisor is of the form $(x-c)$ . Use synthetic division to divide $P(x)=x^3-2x^2-5x+6$ by $(x-3).$ Set up with $c=3$ and coefficients $1,-2,-5,6$ . What are the quotient and remainder (remainder equals $P(3)$ )?

Quotient $x^2+x-2$ , remainder $0$ (correct answer)
Quotient $x^2-5x+10$ , remainder $0$
Quotient $x^2+x-2$ , remainder $6$
Quotient $x^2+x+2$ , remainder $0$

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form

(x - c)

that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is

(x - c)

: instead of the complex long division setup, you just write the value

c

(from

x - c

) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide

x^3 - 2x^2 - 5x + 6

(x - 3)

using synthetic division: identify

c = 3

, write coefficients

1, -2, -5, 6

; bring down

1

, multiply by

3

for

3

, add to

-2

for

1

; multiply

1

3

for

3

, add to

-5

for

-2

; multiply

-2

3

for

-6

, add to

6

for

0

; quotient

x^2 + x - 2

with remainder

0

, and

P(3) = 27 - 18 - 15 + 6 = 0

confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder accurately. Choice C misreads the remainder as

6

: the last bottom row value is the remainder, so verify with direct substitution if unsure. Synthetic division step-by-step: from divisor

(x - c)

, identify

c

; write coefficients in descending order, including zeros if needed; bring down first, multiply by

c

, add to next, repeat; read quotient from bottom row except last, which is remainder. Awesome progress—zero remainder means you've found a factor, perfect for solving polynomials!

Question 19

Use synthetic division to find the quotient and remainder when $P(x)=2x^4+x^3-3x^2+4x-1$ is divided by $(x-2).$ Set up with $c=2$ and coefficients $2,1,-3,4,-1$ . Read the quotient coefficients from the bottom row and the remainder as the last value (which equals $P(2)$ ). Which is correct?

Quotient $2x^3-3x^2+7x+18$ , remainder $35$
Quotient $2x^3+5x^2+7x+18$ , remainder $-35$
Quotient $2x^3+5x^2+7x+18$ , remainder $0$
Quotient $2x^3+5x^2+7x+18$ , remainder $35$ (correct answer)

Explanation: This question tests your understanding of synthetic division—a streamlined shortcut for dividing polynomials by linear divisors of the form (x - c) that's much faster than polynomial long division. Synthetic division is an efficient algorithm for polynomial division when the divisor is (x - c): instead of the complex long division setup, you just write the value c (from x - c) on the left and the polynomial's coefficients on the right, then follow a simple multiply-and-add pattern that produces the quotient coefficients and remainder in one bottom row. To divide 2x^4 + x^3 - 3x^2 + 4x - 1 by (x - 2) using synthetic division: identify c = 2, write coefficients 2, 1, -3, 4, -1; bring down 2, multiply by 2 for 4, add to 1 for 5; multiply 5 by 2 for 10, add to -3 for 7; multiply 7 by 2 for 14, add to 4 for 18; multiply 18 by 2 for 36, add to -1 for 35; quotient 2x^3 + 5x^2 + 7x + 18 with remainder 35, and P(2) = 32 + 8 - 12 + 8 - 1 = 35 confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder accurately. Choice D assumes remainder 0: ensure all additions are precise, especially with higher degrees. Synthetic division step-by-step: from divisor (x - c), identify c; write coefficients in descending order, including zeros if needed; bring down first, multiply by c, add to next, repeat; read quotient from bottom row except last, which is remainder. You're doing wonderfully—this technique shines for higher-degree polynomials!

Question 20

Synthetic division is an efficient method for dividing by $(x-c)$ : write $c$ on the left, list coefficients on the right, then repeatedly bring down, multiply by $c$ , and add. Use synthetic division to divide $P(x)=x^3+2x^2-5x+3$ by $(x-2).$ (Use $c=2$ with coefficients $1,2,-5,3$ .) Read the quotient from the bottom row and the last value as the remainder (which equals $P(2)$ ). Which result is correct?

$Q(x)=x^2+4x+3,\ R=9$ (correct answer)
$Q(x)=x^2+0x-5,\ R=3$
$Q(x)=x^2+4x+3,\ R=-9$
$Q(x)=x^2-4x+3,\ R=9$