Factoring and dividing polynomials

Factoring quadratics and cubics, then dividing one polynomial by another — exactly, with a remainder, and with a number in front of x.

5 sheets · 30 questions · one answer key · 7 pages

Recommended for grades 10–11

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Factoring quadratics Sheet 1 of 5

Factoring polynomialsEasy

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Factor each one. The number in front of x² is not 1.

12x² − x − 1

23x² − 10x − 8

32x² − x − 10

42x² − 3x − 5

53x² − 3

62x² + 9x + 9

Factoring polynomialsEasy

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Factor each one completely. Each has three factors.

1x³ + 3x² − 4x − 12

2x³ + 6x² + 9x + 4

3x³ − 5x² + 7x − 3

4x³ + 8x² + 16x

5x³ − 3x² + 2x

6x³ − 5x² + 6x

Dividing polynomialsExact division

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Divide each polynomial. Every one divides exactly, so every remainder is 0.

1x − 4x³− x²− 14x+ 8

2x + 5x³− x²− 30x

3x − 5x³− 9x²+ 15x+ 25

4x − 2x³− 2x²− 3x+ 6

5x + 4x³+ 5x²+ 8x+ 16

6x − 3x³+ x²− 7x− 15

Dividing polynomialsWith a remainder

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Divide each polynomial. Every one leaves a remainder.

1x + 2x³+ 7x²+ 16x+ 8

2x − 3x³− x²− 2x− 3

3x + 6x³+ 7x²+ 3x− 14

4x + 4x³+ 2x²− 6x+ 7

5x − 5x³− 6x²+ 8x− 19

6x − 3x³+ x²− 7x− 16

Algebra 2Dividing polynomials

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Divide each polynomial. The divisor has a number in front of x.

14x − 14x³− 5x²− 15x+ 4

24x + 34x³− 21x²− 42x− 20

32x + 32x³− x²− 10x− 8

44x + 54x³− 7x²− 15x

53x + 13x³− 14x²+ 4x+ 3

63x − 23x³− 14x²+ 11x− 2

Power Pack · Factoring and dividing polynomialsAnswer key

Factoring quadratics

1. 2x² − x − 1 = (2x + 1)(x − 1)2. 3x² − 10x − 8 = (3x + 2)(x − 4)3. 2x² − x − 10 = (x + 2)(2x − 5)4. 2x² − 3x − 5 = (x + 1)(2x − 5)5. 3x² − 3 = 3(x − 1)(x + 1)6. 2x² + 9x + 9 = (2x + 3)(x + 3)

Factoring cubics

1. x³ + 3x² − 4x − 12 = (x − 2)(x + 2)(x + 3)2. x³ + 6x² + 9x + 4 = (x + 1)²(x + 4)3. x³ − 5x² + 7x − 3 = (x − 1)²(x − 3)4. x³ + 8x² + 16x = x(x + 4)²5. x³ − 3x² + 2x = x(x − 1)(x − 2)6. x³ − 5x² + 6x = x(x − 2)(x − 3)

Dividing exactly

1. (x³ − x² − 14x + 8) ÷ (x − 4) = x² + 3x − 22. (x³ − x² − 30x) ÷ (x + 5) = x² − 6x3. (x³ − 9x² + 15x + 25) ÷ (x − 5) = x² − 4x − 54. (x³ − 2x² − 3x + 6) ÷ (x − 2) = x² − 35. (x³ + 5x² + 8x + 16) ÷ (x + 4) = x² + x + 46. (x³ + x² − 7x − 15) ÷ (x − 3) = x² + 4x + 5

Dividing with a remainder

1. (x³ + 7x² + 16x + 8) ÷ (x + 2) = x² + 5x + 6 remainder −42. (x³ − x² − 2x − 3) ÷ (x − 3) = x² + 2x + 4 remainder 93. (x³ + 7x² + 3x − 14) ÷ (x + 6) = x² + x − 3 remainder 44. (x³ + 2x² − 6x + 7) ÷ (x + 4) = x² − 2x + 2 remainder −15. (x³ − 6x² + 8x − 19) ÷ (x − 5) = x² − x + 3 remainder −46. (x³ + x² − 7x − 16) ÷ (x − 3) = x² + 4x + 5 remainder −1

A number in front of x

1. (4x³ − 5x² − 15x + 4) ÷ (4x − 1) = x² − x − 42. (4x³ − 21x² − 42x − 20) ÷ (4x + 3) = x² − 6x − 6 remainder −23. (2x³ − x² − 10x − 8) ÷ (2x + 3) = x² − 2x − 2 remainder −24. (4x³ − 7x² − 15x) ÷ (4x + 5) = x² − 3x

Power Pack · Factoring and dividing polynomialsAnswer key

A number in front of x

5. (3x³ − 14x² + 4x + 3) ÷ (3x + 1) = x² − 5x + 36. (3x³ − 14x² + 11x − 2) ÷ (3x − 2) = x² − 4x + 1

Factoring and dividing polynomials

What is in the pack

The algebra track

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