Synthetic Division - Algebra 2

1. Synthetic division can evaluate a polynomial quickly because the remainder when dividing by $(x-c)$ equals $P(c)$ (Remainder Theorem). Use synthetic division on $P(x)=x^3-6x^2+11x-6$ with divisor $(x-2)$ (so $c=2$). What is the remainder (i.e., $P(2)$), and is $(x-2)$ a factor?
   Answer: Remainder $ = 0 $; yes, $(x-2)$ is a factor 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 find the remainder for $x^3 - 6x^2 + 11x - 6$ divided by $(x - 2)$ using synthetic division: (1) Identify $c = 2$. (2) Write coefficients: 1, -6, 11, -6. (3) Bring down 1, multiply by 2 for 2, add to -6 for -4; multiply -4 by 2 for -8, add to 11 for 3; multiply 3 by 2 for 6, add to -6 for 0. (4) Remainder 0, so yes a factor, and $P(2) = 8 - 24 + 22 - 6 = 0$ confirms. Choice A correctly executes the synthetic division algorithm and interprets the zero remainder accurately. Choice B might result from misadding the final column, getting 2 instead of 0—always verify with $P(c)$ if the remainder seems off. 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.
2. Use synthetic division (instead of long division) to divide $P(x)=x^4-2x^3+3x^2-4x+5$ by $(x-1)$. Use $c=1$ and coefficients $1,-2,3,-4,5$. Read the quotient from the bottom row (all but the last number) and the remainder from the last number (equal to $P(1)$). Which is correct?
   Answer: Quotient $x^3-x^2+2x-2$, remainder $3$ 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. When dividing a degree-4 polynomial by a linear factor, the quotient will be degree-3, so expect three coefficients plus a remainder in your answer. To divide $x^4 - 2x^3 + 3x^2 - 4x + 5$ by (x - 1): (1) c = 1, coefficients: 1, -2, 3, -4, 5. (2) Bring down 1. (3) 1×1 = 1, add to -2: -2+1 = -1. (4) (-1)×1 = -1, add to 3: 3+(-1) = 2. (5) 2×1 = 2, add to -4: -4+2 = -2. (6) (-2)×1 = -2, add to 5: 5+(-2) = 3. Bottom row: 1, -1, 2, -2, 3, giving quotient $x^3 - x^2 + 2x - 2$ with remainder 3. Choice A correctly identifies the degree-3 quotient $x^3 - x^2 + 2x - 2$ and remainder 3, verified by $P(1) = 1 - 2 + 3 - 4 + 5 = 3$. Other choices might confuse the degree of the quotient—remember it's always one less than the dividend's degree! Synthetic division elegantly handles polynomials of any degree: the process remains the same multiply-and-add pattern, just with more steps. The key is careful bookkeeping and remembering that dividing a degree-n polynomial by (x - c) yields a degree-$(n-1)$ quotient!
3. 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 $=P(1)$.)
   Answer: Quotient $x^2+2x-3$, remainder $-1$ 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!
4. 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?
   Answer: Quotient $x^3-3x^2+3x+2$, remainder $-8$ 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 $x^4 - 2x^3 + 0x^2 + 5x - 6$ by $(x + 1)$ 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 $-3$ by $-1$ for $3$, add to $0$ for $3$; multiply $3$ by $-1$ for $-3$, add to $5$ for $2$; multiply $2$ by $-1$ 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!
5. When a divisor is $(x-c)$, synthetic division is a streamlined alternative to long division: list coefficients (use $0$ if needed), then bring down, multiply by $c$, and add. Use synthetic division to divide $P(x)=x^4-2x^3+0x^2+5x-1$ by $(x+1)=(x-(-1))$ (so $c=-1$). What are the quotient and remainder (remainder $=P(-1)$)?
   Answer: Quotient $x^3-3x^2+3x+2$, remainder $-3$ 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 $x^4 - 2x^3 + 0x^2 + 5x - 1$ by $(x + 1)$ using synthetic division: (1) Identify $c = -1$. (2) Write coefficients: 1, -2, 0, 5, -1. (3) Bring down 1, multiply by -1 for -1, add to -2 for -3; multiply -3 by -1 for 3, add to 0 for 3; multiply 3 by -1 for -3, add to 5 for 2; multiply 2 by -1 for -2, add to -1 for -3. (4) Bottom row: 1, -3, 3, 2, -3, so quotient $x^3 - 3x^2 + 3x + 2$ with remainder -3, and $P(-1) = 1 + 2 - 5 - 1 = -3$ confirms. Choice A correctly executes the synthetic division algorithm and reads the quotient and remainder from the bottom row accurately. Choice B flips the sign of the remainder, perhaps from a final addition error—remember to handle negative multiplications carefully. 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.
6. Use synthetic division (more efficient than long division for $(x-c)$) to divide $P(x)=3x^3-2x^2-7x+5$ by $(x-2).$ What are the quotient and remainder? (Remainder should match $P(2)$.)
   Answer: Quotient $3x^2+4x+1$, remainder $7$ 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 3x³ - 2x² - 7x + 5 by (x - 2) using synthetic division: (1) Identify c = 2 from divisor (x - 2). (2) Write coefficients: 3, -2, -7, 5. (3) Setup: 2 on left, coefficients on top. (4) Bring down 3. (5) Multiply 3 × 2 = 6, write under -2, add: -2 + 6 = 4. (6) Multiply 4 × 2 = 8, write under -7, add: -7 + 8 = 1. (7) Multiply 1 × 2 = 2, write under 5, add: 5 + 2 = 7. Bottom row: 3, 4, 1, 7. This means quotient is 3x² + 4x + 1 with remainder 7. The Remainder Theorem confirms: P(2) = 24 - 8 - 14 + 5 = 7, yes! Choice A correctly identifies the quotient as 3x² + 4x + 1 with remainder 7. Choice B's error of remainder 0 would mean (x - 2) is a factor, but P(2) = 7 ≠ 0 proves it's not! Synthetic division step-by-step: the leading coefficient (3 here) always comes down unchanged as the first coefficient of the quotient. Why synthetic division is worth learning: it handles any polynomial coefficients (not just 1) with the same simple algorithm!
7. 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 $=P(3)$.)
   Answer: Quotient $x^2+x-2$, remainder $0$ 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!
8. 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?
   Answer: Quotient $x^3-3x^2+3x+2$, remainder $-6$ 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!
9. 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?
   Answer: Quotient $2x^2-x+1$, remainder $0$ 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 divides polynomials: for $2x^3 - 7x^2 + 4x - 3$ 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!
10. 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.
    Answer: Quotient $2x^3+x^2+4x-1$, remainder $5$ 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 $2x^4 - x^3 + 3x^2 - 5x + 6$ by $(x - 1)$ 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.
11. 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?
    Answer: $Q(x)=x^2+2x-3,\ R=-1$ 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. 1*1=1, 1+1=2. 2*1=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.
12. 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 $k$?
    Answer: $k = 1$, which matches the remainder from the division 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.
13. Divide $P(x)=x^3+x^2-5x+2$ by $(x-1)$ using synthetic division (more efficient than long division for $(x-c)$). Use $c=1$ and coefficients $1,1,-5,2$. Read the quotient coefficients from the bottom row and the remainder from the last entry (which equals $P(1)$). What is the quotient and remainder?
    Answer: Quotient $x^2+2x-3$, remainder $-1$ 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: write $c = 1$ (from x - 1) on the left and coefficients 1, 1, -5, 2 on the right, then follow the multiply-and-add pattern to get quotient and remainder in one bottom row. To divide $x^3 + x^2 - 5x + 2$ by $(x - 1)$): (1) Bring down $1$. (2) $1\times1 = 1$, add to $1$: $1+1 = 2$. (3) $2\times1 = 2$, add to $-5$: $-5+2 = -3$. (4) $(-3)\times1 = -3$, add to $2$: $2+(-3) = -1$. Bottom row: $1, 2, -3, -1$, giving quotient $x^2 + 2x - 3$ with remainder $-1$. Choice A correctly executes the synthetic division algorithm and reads the quotient coefficients ($1, 2, -3$) and remainder (-1) from the bottom row. Other choices have arithmetic errors or misread the results—remember to carefully track signs during the multiply-and-add steps! The Remainder Theorem confirms our answer: $P(1) = 1 + 1 - 5 + 2 = -1$, matching our remainder. Synthetic division's power lies in its simplicity: one algorithm gives both quotient and remainder, making it invaluable for polynomial division, evaluation, and factoring!
14. 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 $=P(-3)$.)
    Answer: $39$ 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!
15. 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?
    Answer: Quotient $x^2+x-2$, remainder $0$ 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 $x^3 - 2x^2 - 5x + 6$ by $(x - 3)$: (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!
16. Synthetic division can evaluate a polynomial efficiently: when dividing by $(x-c)$, the remainder equals $P(c)$ (Remainder Theorem). Use synthetic division with $c=2$ to evaluate $P(x)=x^3-x^2-4x+4$ by finding the remainder when $P(x)$ is divided by $(x-2)$. What is $P(2)$?
    Answer: $0$ 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. The Remainder Theorem states that when P(x) is divided by (x - c), the remainder equals P(c), making synthetic division a clever way to evaluate polynomials: just find the remainder! To evaluate P(2) for x³ - x² - 4x + 4: (1) Set up with c = 2 and coefficients 1, -1, -4, 4. (2) Bring down 1. (3) 1×2 = 2, add to -1: -1+2 = 1. (4) 1×2 = 2, add to -4: -4+2 = -2. (5) (-2)×2 = -4, add to 4: 4+(-4) = 0. Bottom row: 1, 1, -2, 0, so remainder = 0 = P(2). Choice A correctly identifies P(2) = 0, which we can verify directly: P(2) = 8 - 4 - 8 + 4 = 0. The zero remainder reveals that x = 2 is a zero of P(x), meaning (x - 2) is a factor! Synthetic division serves triple duty: it divides polynomials, evaluates them via the Remainder Theorem, and identifies zeros/factors when remainder = 0. This makes it an essential tool for analyzing polynomials—much faster than substituting values directly into the original polynomial!
17. 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?
    Answer: $39$ 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!
18. Synthetic division works efficiently for divisors of the form $(x-c)$. Use synthetic division to test whether $(x-2)$ is a factor of $P(x)=x^3-3x^2+2x-8.$ If the remainder is $0$, then $(x-2)$ is a factor (Remainder Theorem). What happens?
    Answer: No; remainder $-8$ 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) is a factor of x³ - 3x² + 2x - 8 using synthetic division: (1) Identify c = 2 from divisor (x - 2). (2) Write coefficients: 1, -3, 2, -8. (3) Setup: 2 on left, coefficients on top. (4) Bring down 1. (5) Multiply 1 × 2 = 2, write under -3, add: -3 + 2 = -1. (6) Multiply -1 × 2 = -2, write under 2, add: 2 + (-2) = 0. (7) Multiply 0 × 2 = 0, write under -8, add: -8 + 0 = -8. Bottom row: 1, -1, 0, -8. The remainder is -8, not 0, so (x - 2) is NOT a factor. The Remainder Theorem confirms: P(2) = 8 - 12 + 4 - 8 = -8, yes! Choice B correctly identifies that (x - 2) is not a factor with remainder -8. Choice D's error of remainder 8 suggests a sign mistake—be careful with negative coefficients! Synthetic division step-by-step: a remainder of 0 means the divisor is a factor, while any non-zero remainder means it's not a factor. Why synthetic division is worth learning: it makes factor testing incredibly fast—you can quickly check multiple potential factors when combined with the Rational Zeros Theorem!
19. 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?
    Answer: $Q(x)=x^2+4x+3,\ R=9$ 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 +3 by (x-2): c=2, coefficients 1,2,-5,3. Bring down 1. 1*2=2, 2+2=4. 4*2=8, -5+8=3. 3*2=6, 3+6=9. Bottom: 1,4,3,9. Q=$x^2$ +4x +3, R=9. Matches A. Choice C has R=-9, likely sign error in last add. You're doing great learning this efficient tool!
20. A quick way to divide a polynomial by a linear divisor $(x-c)$ is synthetic division, which is usually faster than long division. Use synthetic division to divide $P(x)=x^3+4x^2-3x+7$ by $(x-2).$ (Use $c=2$ on the left and the coefficients $1,4,-3,7$ on the top. The bottom row gives the quotient coefficients, and the last value is the remainder, which also equals $P(2)$ by the Remainder Theorem.) What are the quotient and remainder? Write your result in the form $P(x)=(x-2)Q(x)+R$.
    Answer: $Q(x)=x^2+6x+9,\ R=25$ 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 $x^3 + 4x^2 - 3x + 7$ by $(x - 2)$ using synthetic division: (1) Identify c = 2 from divisor $(x - 2)$. (2) Write coefficients: 1, 4, -3, 7. (3) Bring down 1. (4) Multiply 1 times 2 = 2, add to 4: 6. (5) Multiply 6 times 2 = 12, add to -3: 9. (6) Multiply 9 times 2 = 18, add to 7: 25. (7) Bottom row: 1, 6, 9, 25. This means quotient is $x^2 + 6x + 9$ with remainder 25. The Remainder Theorem confirms: $P(2) = 8 + 16 - 6 + 7 = 25$, perfect! Choice B correctly executes the synthetic division algorithm and reads the quotient and remainder from the bottom row accurately. Choice A makes an arithmetic error during the multiply-and-add steps: synthetic division requires careful calculation at each step because errors propagate—if you get one value wrong, all subsequent values will be wrong too, so double-check each multiplication and addition before moving to the next column. 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—when checking if $x = c$ is a zero, synthetic division gives the remainder immediately (if 0, it's a zero!) plus the quotient for further factoring.

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