AI Tutor
Back to College Algebra

1.5 Factoring Polynomials

Pulling out a greatest common factor, trinomials, grouping, perfect squares, differences of squares, sums and differences of cubes, and negative-exponent expressions.

Algebra and Trigonometry 2e, 1.5 Factoring Polynomials

Status
FlagsLoading your progress…

46 exercises

Factoring the Greatest Common Factor

Factor 6 x 3 y 3 +45 x 2 y 2 +21xy. 6 x^{3} y^{3} + 45 x^{2} y^{2} + 21 x y .

2.5

1.5 Exercise 5

Factor x( b 2 −a)+6( b 2 −a) x \left(b^{2} - a\right) + 6 \left(b^{2} - a\right) by pulling out the GCF.

2.5

Factoring a Trinomial with Leading Coefficient 1

Factor x 2 +2x−15. x^{2} + 2 x - 15.

2.5

1.5 Exercise 7

Factor x 2 −7x+6. x^{2} - 7 x + 6.

2.5

Factoring a Trinomial by Grouping

Factor 5 x 2 +7x−6 5 x^{2} + 7 x - 6 by grouping.

2.5

1.5 Exercise 9

Factor

  1. ⓐ 2 x 2 +9x+9 2 x^{2} + 9 x + 9
  2. ⓑ 6 x 2 +x−1 6 x^{2} + x - 1

2.5

Factoring a Perfect Square Trinomial

Factor 25 x 2 +20x+4. 25 x^{2} + 20 x + 4.

2.5

1.5 Exercise 11

Factor 49 x 2 −14x+1. 49 x^{2} - 14 x + 1.

2.5

Factoring a Difference of Squares

Factor 9 x 2 −25. 9 x^{2} - 25.

2.5

1.5 Exercise 13

Factor 81 y 2 −100. 81 y^{2} - 100.

2.5

Factoring a Sum of Cubes

Factor x 3 +512. x^{3} + 512.

2.5

1.5 Exercise 15

Factor the sum of cubes: 216 a 3 + b 3 . 216 a^{3} + b^{3} .

2.5

Factoring a Difference of Cubes

Factor 8 x 3 −125. 8 x^{3} - 125.

2.5

1.5 Exercise 17

Factor the difference of cubes: 1,000 x 3 −1. 1,000 x^{3} - 1.

2.5

Factoring an Expression with Fractional or Negative Exponents

Factor 3x (x+2) −1 3 +4 (x+2) 2 3 . 3 x \left(x + 2\right)^{\frac{−1}{3}} + 4 \left(x + 2\right)^{\frac{2}{3}} .

2.5

1.5 Exercise 19

Factor 2 (5a−1) 3 4 +7a (5a−1) − 1 4 . 2 \left(5 a - 1\right)^{\frac{3}{4}} + 7 a \left(5 a - 1\right)^{- \frac{1}{4}} .

2.5

1.5 Exercise 20

If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.

2.5

1.5 Exercise 22

How do you factor by grouping?

2.5

1.5 Exercise 24

Find the greatest common factor.

49m b 2 −35 m 2 ba+77m a 2 49 m b^{2} - 35 m^{2} b a + 77 m a^{2}

2.5

1.5 Exercise 26

Find the greatest common factor.

200 p 3 m 3 −30 p 2 m 3 +40 m 3 200 p^{3} m^{3} - 30 p^{2} m^{3} + 40 m^{3}

2.5

1.5 Exercise 28

Find the greatest common factor.

6 y 4 −2 y 3 +3 y 2 −y 6 y^{4} - 2 y^{3} + 3 y^{2} - y

2.5

1.5 Exercise 30

Factor by grouping.

2 a 2 +9a−18 2 a^{2} + 9 a - 18

2.5

1.5 Exercise 32

Factor by grouping.

6 n 2 −19n−11 6 n^{2} - 19 n - 11

2.5

1.5 Exercise 34

Factor by grouping.

2 p 2 −5p−7 2 p^{2} - 5 p - 7

2.5

1.5 Exercise 36

Factor the polynomial.

10 h 2 −9h−9 10 h^{2} - 9 h - 9

2.5

1.5 Exercise 38

Factor the polynomial.

9 d 2 −73d+8 9 d^{2} −73 d + 8

2.5

1.5 Exercise 40

Factor the polynomial.

12 t 2 +t−13 12 t^{2} + t - 13

2.5

1.5 Exercise 42

Factor the polynomial.

16 x 2 −100 16 x^{2} - 100

2.5

1.5 Exercise 44

Factor the polynomial.

121 p 2 −169 121 p^{2} - 169

2.5

1.5 Exercise 46

Factor the polynomial.

361 d 2 −81 361 d^{2} - 81

2.5

1.5 Exercise 48

Factor the polynomial.

144 b 2 −25 c 2 144 b^{2} - 25 c^{2}

2.5

1.5 Exercise 50

Factor the polynomial.

49 n 2 +168n+144 49 n^{2} + 168 n + 144

2.5

1.5 Exercise 52

Factor the polynomial.

225 y 2 +120y+16 225 y^{2} + 120 y + 16

2.5

1.5 Exercise 54

Factor the polynomial.

25 p 2 −120p+144 25 p^{2} - 120 p + 144

2.5

1.5 Exercise 56

Factor the polynomials.

x 3 +216 x^{3} + 216

2.5

1.5 Exercise 58

Factor the polynomials.

125 a 3 +343 125 a^{3} + 343

2.5

1.5 Exercise 60

Factor the polynomials.

64 x 3 −125 64 x^{3} −125

2.5

1.5 Exercise 62

Factor the polynomials.

125 r 3 +1,728 s 3 125 r^{3} + 1,728 s^{3}

2.5

1.5 Exercise 64

Factor the polynomials.

3c ( 2c+3 ) − 1 4 −5 ( 2c+3 ) 3 4 3 c \left(2 c + 3\right)^{- \frac{1}{4}} - 5 \left(2 c + 3\right)^{\frac{3}{4}}

2.5

1.5 Exercise 66

Factor the polynomials.

14x ( x+2 ) − 2 5 +5 ( x+2 ) 3 5 14 x \left(x + 2\right)^{- \frac{2}{5}} + 5 \left(x + 2\right)^{\frac{3}{5}}

2.5

1.5 Exercise 68

Factor the polynomials.

5z (2z−9) − 3 2 +11 (2z−9) − 1 2 5 z \left(2 z - 9\right)^{- \frac{3}{2}} + 11 \left(2 z - 9\right)^{- \frac{1}{2}}

2.5

1.5 Exercise 70

Factor by grouping to find the length and width of the park.

2.5

1.5 Exercise 72

At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is 9 x 2 −25  m 2 . 9 x^{2} - 25 \textrm{ } \text{m}^{2} . Factor the area to find the lengths of the sides of the fountain.

2.5

1.5 Exercise 74

Factor the polynomials completely.

16 x 4 −200 x 2 +625 16 x^{4} - 200 x^{2} + 625

2.5

1.5 Exercise 76

Factor the polynomials completely.

16 z 4 −2,401 a 4 16 z^{4} - 2,401 a^{4}

2.5

1.5 Exercise 78

Factor the polynomials completely.

(32 x 3 +48 x 2 −162x−243) −1 \left(32 x^{3} + 48 x^{2} - 162 x - 243\right)^{−1}

2.5

Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.