"

Chapter 7: Factoring

7.1 Greatest Common Factor

The opposite of multiplying polynomials together is factoring polynomials. Factored polynomials help to solve equations, learn behaviours of graphs, work with fractions and more. Because so many concepts in algebra depend on us being able to factor polynomials, it is important to have very strong factoring skills.

In this section, the focus is on factoring using the greatest common factor or GCF of a polynomial. When you previously multiplied polynomials, you multiplied monomials by polynomials by distributing, solving problems such as 4x2(2x23x+8) to yield 8x412x3+32x2. For factoring, you will work the same problem backwards. For instance, you could start with the polynomial 8x412x3+32x2 and work backwards to 4x2(2x23x+8).

To do this, first identify the GCF of a polynomial. Look at finding the GCF of several numbers. To find the GCF of several numbers, look for the largest number that each of the numbers can be divided by.

Example 7.1.1

Find the GCF of 15, 24, 27.

First, break all these numbers into their primes.

15=3×524=2×2×2×3 or 23×327=3×3×3 or 33

By observation, the only number that each can be divided by is 3. Therefore, the GCF = 3.

Example 7.1.2

Find the GCF of 24x4y2z, 18x2y4, and 12x3yz5.

First, break all these numbers into their primes. (Use • to designate multiplication instead of ×.)

24x4y2z=233x4y2z18x2y4=232x2y412x3yz5=223x3yz5

By observation, what is shared between all three monomials is 23x2y or 6x2y.

Questions

Factor out the common factor in each of the following polynomials.

  1. 9+8b2
  2. x5
  3. 45x225
  4. 1+2n2
  5. 5635p
  6. 50x80y
  7. 7ab35a2b
  8. 27x2y572x3y2
  9. 3a2b+6a3b2
  10. 8x3y2+4x3
  11. 5x25x315x4
  12. 32n9+32n6+40n5
  13. 28m4+40m3+8
  14. 10x4+20x2+12x
  15. 30b9+5ab15a2
  16. 27y7+12y2x+9y2
  17. 48a2b256a3b56a5b
  18. 30m6+15mn225
  19. 20x8y2z2+15x5y2z+35x3y3z
  20. 3p+12q15q2r2
  21. 18n5+3n321n+3
  22. 30a8+6a5+27a3+21a2
  23. 40x1120x12+50x1350x14
  24. 24x64x4+12x3+4x2
  25. 32mn8+4m6n+12mn4+16mn
  26. 10y7+6y104y10x8y8x

Answer Key 7.1

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book