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Remainder & Factor Theorems — Easy Worksheet 2
Grade 11 · MathSheets Name: ____________________ Date: __________
Use synthetic division or the Remainder and Factor Theorems.
How to solve
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Example
Find the remainder when f(x) = -x³ - 2x² - 2x + 7 is divided by (x + 3).
| -3 | -1 | -2 | -2 | 7 |
Answer: 22
| 2 | 2 | 3 | -1 | 3 |
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients 2, 3, -1, 3: Bring down 2. 2 × 2 = 4, and 3 + 4 = 7. 7 × 2 = 14, and -1 + 14 = 13. 13 × 2 = 26, and 3 + 26 = 29. The last number is the remainder: 29.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients 2, 3, -1, 3: Bring down 2. 2 × 2 = 4, and 3 + 4 = 7. 7 × 2 = 14, and -1 + 14 = 13. 13 × 2 = 26, and 3 + 26 = 29. The last number is the remainder: 29.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = x³ + 3x² - x + 2 is divided by (x + 2).
| -2 | 1 | 3 | -1 | 2 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(3). Synthetic division by 3 with the coefficients 2, 3, 3, 2: Bring down 2. 2 × 3 = 6, and 3 + 6 = 9. 9 × 3 = 27, and 3 + 27 = 30. 30 × 3 = 90, and 2 + 90 = 92. The last number is the remainder: 92.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(3). Synthetic division by 3 with the coefficients 2, 3, 3, 2: Bring down 2. 2 × 3 = 6, and 3 + 6 = 9. 9 × 3 = 27, and 3 + 27 = 30. 30 × 3 = 90, and 2 + 90 = 92. The last number is the remainder: 92.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = x³ + x² + 2x + 1 is divided by (x + 1).
| -1 | 1 | 1 | 2 | 1 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 By the Factor Theorem, (x + 2) is a factor exactly when f(-2) = 0. Synthetic division by -2 with the coefficients 1, 1, 0, -1: Bring down 1. 1 × (-2) = -2, and 1 + (-2) = -1. -1 × (-2) = 2, and 0 + 2 = 2. 2 × (-2) = -4, and -1 + (-4) = -5. The remainder is -5, not 0, so the answer is no.
✓ Correct!
Not quite. The answer is
💡 By the Factor Theorem, (x + 2) is a factor exactly when f(-2) = 0. Synthetic division by -2 with the coefficients 1, 1, 0, -1: Bring down 1. 1 × (-2) = -2, and 1 + (-2) = -1. -1 × (-2) = 2, and 0 + 2 = 2. 2 × (-2) = -4, and -1 + (-4) = -5. The remainder is -5, not 0, so the answer is no.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = -x³ + 3x² + x + 3 is divided by (x - 3).
| 3 | -1 | 3 | 1 | 3 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
| -1 | -1 | 3 | -1 | -3 |
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-1). Synthetic division by -1 with the coefficients -1, 3, -1, -3: Bring down -1. -1 × (-1) = 1, and 3 + 1 = 4. 4 × (-1) = -4, and -1 + (-4) = -5. -5 × (-1) = 5, and -3 + 5 = 2. The last number is the remainder: 2.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-1). Synthetic division by -1 with the coefficients -1, 3, -1, -3: Bring down -1. -1 × (-1) = 1, and 3 + 1 = 4. 4 × (-1) = -4, and -1 + (-4) = -5. -5 × (-1) = 5, and -3 + 5 = 2. The last number is the remainder: 2.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = 3x³ - 3x² - 2x + 1 is divided by (x - 1).
| 1 | 3 | -3 | -2 | 1 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-3). Synthetic division by -3 with the coefficients 2, -1, 2, 7: Bring down 2. 2 × (-3) = -6, and -1 + (-6) = -7. -7 × (-3) = 21, and 2 + 21 = 23. 23 × (-3) = -69, and 7 + (-69) = -62. The last number is the remainder: -62.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-3). Synthetic division by -3 with the coefficients 2, -1, 2, 7: Bring down 2. 2 × (-3) = -6, and -1 + (-6) = -7. -7 × (-3) = 21, and 2 + 21 = 23. 23 × (-3) = -69, and 7 + (-69) = -62. The last number is the remainder: -62.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = x³ - x² + 3x - 3 is divided by (x + 2).
| -2 | 1 | -1 | 3 | -3 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 By the Factor Theorem, (x - 2) is a factor exactly when f(2) = 0. Synthetic division by 2 with the coefficients 1, 1, -2, -8: Bring down 1. 1 × 2 = 2, and 1 + 2 = 3. 3 × 2 = 6, and -2 + 6 = 4. 4 × 2 = 8, and -8 + 8 = 0. The remainder is 0, so the answer is yes.
✓ Correct!
Not quite. The answer is
💡 By the Factor Theorem, (x - 2) is a factor exactly when f(2) = 0. Synthetic division by 2 with the coefficients 1, 1, -2, -8: Bring down 1. 1 × 2 = 2, and 1 + 2 = 3. 3 × 2 = 6, and -2 + 6 = 4. 4 × 2 = 8, and -8 + 8 = 0. The remainder is 0, so the answer is yes.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = x³ + x² + 1 is divided by (x - 2).
| 2 | 1 | 1 | 0 | 1 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients -1, 2, 1, -1: Bring down -1. -1 × 2 = -2, and 2 + (-2) = 0. 0 × 2 = 0, and 1 + 0 = 1. 1 × 2 = 2, and -1 + 2 = 1. The last number is the remainder: 1.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients -1, 2, 1, -1: Bring down -1. -1 × 2 = -2, and 2 + (-2) = 0. 0 × 2 = 0, and 1 + 0 = 1. 1 × 2 = 2, and -1 + 2 = 1. The last number is the remainder: 1.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = -x³ + 2x² + 2x + 3 is divided by (x - 2).
| 2 | -1 | 2 | 2 | 3 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 By the Factor Theorem, (x - 3) is a factor exactly when f(3) = 0. Synthetic division by 3 with the coefficients -1, 1, -2, 24: Bring down -1. -1 × 3 = -3, and 1 + (-3) = -2. -2 × 3 = -6, and -2 + (-6) = -8. -8 × 3 = -24, and 24 + (-24) = 0. The remainder is 0, so the answer is yes.
✓ Correct!
Not quite. The answer is
💡 By the Factor Theorem, (x - 3) is a factor exactly when f(3) = 0. Synthetic division by 3 with the coefficients -1, 1, -2, 24: Bring down -1. -1 × 3 = -3, and 1 + (-3) = -2. -2 × 3 = -6, and -2 + (-6) = -8. -8 × 3 = -24, and 24 + (-24) = 0. The remainder is 0, so the answer is yes.
💡 How to solve: Remainder & Factor Theorems
Remainder Theorem: the remainder of f(x) ÷ (x - c) equals f(c), so synthetic division by c (or substituting c) gives it. Factor Theorem: (x - c) is a factor exactly when f(c) = 0. When it is, the quotient from synthetic division holds the other factors.
Try a similar one
Find the remainder when f(x) = x³ - 3x² - 2x - 5 is divided by (x - 2).
| 2 | 1 | -3 | -2 | -5 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Remainder & Factor Theorems — Easy Worksheet 2
Use synthetic division or the Remainder and Factor Theorems.