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 $x^3 + x^2 - 5x + 2$ by $(x - 1)$ 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.

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!

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!

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!

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 $x^3 + 2x^2 - x - 2$ by $(x + 1)$: (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!

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!

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!

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!

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.

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!