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Remainder & Factor Theorems — Easy Worksheet 1
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
| -1 | -1 | 2 | -3 | -3 |
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-1). Synthetic division by -1 with the coefficients -1, 2, -3, -3: Bring down -1. -1 × (-1) = 1, and 2 + 1 = 3. 3 × (-1) = -3, and -3 + (-3) = -6. -6 × (-1) = 6, and -3 + 6 = 3. The last number is the remainder: 3.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-1). Synthetic division by -1 with the coefficients -1, 2, -3, -3: Bring down -1. -1 × (-1) = 1, and 2 + 1 = 3. 3 × (-1) = -3, and -3 + (-3) = -6. -6 × (-1) = 6, and -3 + 6 = 3. The last number is the remainder: 3.
💡 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 - x² + x + 3 is divided by (x + 3).
| -3 | 3 | -1 | 1 | 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 Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients -1, 1, -3, 2: Bring down -1. -1 × 2 = -2, and 1 + (-2) = -1. -1 × 2 = -2, and -3 + (-2) = -5. -5 × 2 = -10, and 2 + (-10) = -8. The last number is the remainder: -8.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients -1, 1, -3, 2: Bring down -1. -1 × 2 = -2, and 1 + (-2) = -1. -1 × 2 = -2, and -3 + (-2) = -5. -5 × 2 = -10, and 2 + (-10) = -8. The last number is the remainder: -8.
💡 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 - x - 2 is divided by (x + 3).
| -3 | 1 | -2 | -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 Factor Theorem, (x + 3) is a factor exactly when f(-3) = 0. Synthetic division by -3 with the coefficients 2, -3, 3, 90: Bring down 2. 2 × (-3) = -6, and -3 + (-6) = -9. -9 × (-3) = 27, and 3 + 27 = 30. 30 × (-3) = -90, and 90 + (-90) = 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 2, -3, 3, 90: Bring down 2. 2 × (-3) = -6, and -3 + (-6) = -9. -9 × (-3) = 27, and 3 + 27 = 30. 30 × (-3) = -90, and 90 + (-90) = 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) = 2x³ - 3x² - 2x - 5 is divided by (x - 1).
| 1 | 2 | -3 | -2 | -5 |
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
| 2 | 1 | -3 | -1 | 1 |
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients 1, -3, -1, 1: Bring down 1. 1 × 2 = 2, and -3 + 2 = -1. -1 × 2 = -2, and -1 + (-2) = -3. -3 × 2 = -6, and 1 + (-6) = -5. The last number is the remainder: -5.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(2). Synthetic division by 2 with the coefficients 1, -3, -1, 1: Bring down 1. 1 × 2 = 2, and -3 + 2 = -1. -1 × 2 = -2, and -1 + (-2) = -3. -3 × 2 = -6, and 1 + (-6) = -5. The last number is the remainder: -5.
💡 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 - 1 is divided by (x - 2).
| 2 | -1 | -2 | 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, -2: Bring down -1. -1 × (-2) = 2, and 2 + 2 = 4. 4 × (-2) = -8, and 1 + (-8) = -7. -7 × (-2) = 14, and -2 + 14 = 12. The last number is the remainder: 12.
✓ 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, -2: Bring down -1. -1 × (-2) = 2, and 2 + 2 = 4. 4 × (-2) = -8, and 1 + (-8) = -7. -7 × (-2) = 14, and -2 + 14 = 12. The last number is the remainder: 12.
💡 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 - 3 is divided by (x + 2).
| -2 | 3 | -3 | 0 | -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) = -1. -1 × (-2) = 2, and 2 + 2 = 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) = -1. -1 × (-2) = 2, and 2 + 2 = 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) = 2x³ + x² - x - 1 is divided by (x - 2).
| 2 | 2 | 1 | -1 | -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(-1). Synthetic division by -1 with the coefficients 3, 0, -3, 3: Bring down 3. 3 × (-1) = -3, and 0 + (-3) = -3. -3 × (-1) = 3, and -3 + 3 = 0. 0 × (-1) = 0, and 3 + 0 = 3. The last number is the remainder: 3.
✓ Correct!
Not quite. The answer is
💡 By the Remainder Theorem the remainder is f(-1). Synthetic division by -1 with the coefficients 3, 0, -3, 3: Bring down 3. 3 × (-1) = -3, and 0 + (-3) = -3. -3 × (-1) = 3, and -3 + 3 = 0. 0 × (-1) = 0, and 3 + 0 = 3. The last number is the remainder: 3.
💡 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 - 1 is divided by (x - 1).
| 1 | 1 | 3 | 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 Factor Theorem, (x - 2) is a factor exactly when f(2) = 0. Synthetic division by 2 with the coefficients 1, -2, -3, 6: Bring down 1. 1 × 2 = 2, and -2 + 2 = 0. 0 × 2 = 0, and -3 + 0 = -3. -3 × 2 = -6, and 6 + (-6) = 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, -2, -3, 6: Bring down 1. 1 × 2 = 2, and -2 + 2 = 0. 0 × 2 = 0, and -3 + 0 = -3. -3 × 2 = -6, and 6 + (-6) = 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 - 1 is divided by (x + 3).
| -3 | 1 | 3 | 2 | -1 |
✓ 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 1
Use synthetic division or the Remainder and Factor Theorems.