Synthetic Division

Use synthetic division for polynomial division

Synthetic division content

📚 Practice Problems

1Problem 1easy

❓ Question:

Use synthetic division to divide (x³ + 4x² - 7x + 2) by (x - 1).

💡 Show Solution

Step 1: Set up synthetic division with a = 1: 1 | 1 4 -7 2 |

Step 2: Bring down the first coefficient: 1 | 1 4 -7 2 |
_______________ 1

Step 3: Multiply and add repeatedly: 1 × 1 = 1, then 4 + 1 = 5 1 × 5 = 5, then -7 + 5 = -2 1 × (-2) = -2, then 2 + (-2) = 0

1 |  1   4  -7   2
  |      1   5  -2
  _______________
     1   5  -2   0

Step 4: Write the result: Quotient: x² + 5x - 2 Remainder: 0

Answer: x² + 5x - 2

2Problem 2easy

❓ Question:

Use synthetic division to find the remainder when (2x⁴ - 3x² + 5) is divided by (x + 2).

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Step 1: Rewrite with all terms: 2x⁴ + 0x³ - 3x² + 0x + 5

Step 2: For (x + 2), use a = -2: -2 | 2 0 -3 0 5 |

Step 3: Perform synthetic division: -2 | 2 0 -3 0 5 | -4 8 -10 20 _____________________ 2 -4 5 -10 25

Step 4: The remainder is the last number: Remainder = 25

Step 5: Verify using Remainder Theorem: P(-2) = 2(-2)⁴ - 3(-2)² + 5 = 2(16) - 3(4) + 5 = 32 - 12 + 5 = 25 ✓

Answer: Remainder = 25

3Problem 3medium

❓ Question:

Given that (x - 3) is a factor of P(x) = x³ - 2x² - 9x + k, find the value of k.

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Step 1: Use the Factor Theorem: If (x - 3) is a factor, then P(3) = 0

Step 2: Substitute x = 3: P(3) = (3)³ - 2(3)² - 9(3) + k = 0 = 27 - 18 - 27 + k = 0 = -18 + k = 0 = k = 18

Step 3: Verify using synthetic division: 3 | 1 -2 -9 18 | 3 3 -18 __________________ 1 1 -6 0 ✓

Remainder is 0, confirming (x - 3) is a factor

Answer: k = 18

4Problem 4medium

❓ Question:

Use synthetic division to completely factor P(x) = x³ - 6x² + 11x - 6, given that x = 1 is a zero.

💡 Show Solution

Step 1: Use synthetic division with a = 1: 1 | 1 -6 11 -6 | 1 -5 6 __________________ 1 -5 6 0

Quotient: x² - 5x + 6

Step 2: Factor the quotient: x² - 5x + 6 = (x - 2)(x - 3)

Step 3: Write complete factorization: P(x) = (x - 1)(x - 2)(x - 3)

Step 4: Find all zeros: x = 1, 2, 3

Step 5: Verify by expanding: (x - 1)(x - 2)(x - 3) = (x - 1)(x² - 5x + 6) = x³ - 5x² + 6x - x² + 5x - 6 = x³ - 6x² + 11x - 6 ✓

Answer: P(x) = (x - 1)(x - 2)(x - 3), zeros: 1, 2, 3

5Problem 5hard

❓ Question:

Use synthetic division twice to find all zeros of P(x) = 2x⁴ - x³ - 18x² + 9x, given that x = 3 and x = -3 are zeros.

💡 Show Solution

Step 1: Factor out common x: P(x) = x(2x³ - x² - 18x + 9)

Step 2: Use synthetic division on 2x³ - x² - 18x + 9 with a = 3: 3 | 2 -1 -18 9 | 6 15 -9 _________________ 2 5 -3 0

Result: 2x² + 5x - 3

Step 3: Use synthetic division on 2x² + 5x - 3 with a = -3: -3 | 2 5 -3 | -6 3 _____________ 2 -1 0

Result: 2x - 1

Step 4: Write complete factorization: P(x) = x(x - 3)(x + 3)(2x - 1)

Step 5: Find all zeros: x = 0 x = 3 x = -3 x = 1/2

Step 6: Verify the count: Degree 4 polynomial has 4 zeros ✓

Answer: Zeros are x = 0, 3, -3, 1/2 Factored form: P(x) = x(x - 3)(x + 3)(2x - 1)

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