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!

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 $x^3 - 2x^2 - 5x + 6$ by $(x - 3)$ 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 $1$ by $3$ for $3$, add to $-5$ for $-2$; multiply $-2$ by $3$ 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!

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!

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!

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.

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. 12=2, 2+2=4. 42=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!

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 $x^3 -6x^2 +11x -6$ by $(x-3)$: $c=3$, coefficients $1,-6,11,-6$. Bring down $1$. $13=3$, $-6+3=-3$. $-33=-9$, $11+-9=2$. $2*3=6$, $-6+6=0$. Bottom: $1,-3,2,0$. $Q(x)=x^2 -3x +2$, $R=0$. Matches A. Choice D has positive signs, perhaps used c=-3 wrongly. Keep up the great work mastering this!

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-2): c=2, coefficients 1,-2,0,5,-6. Bring down 1. 12=2, -2+2=0. 02=0, 0+0=0. 02=0, 5+0=5. 52=10, -6+10=4. Bottom: 1,0,0,5,4. $Q=x^3$ $+0x^2$ +0x +5, R=4. Matches A. Choice B has R=-6, perhaps misadded last column. Synthetic division step-by-step: always include 0 for missing terms to keep degrees aligned. This technique speeds up polynomial work tremendously!

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.

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.