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 4x^2(2x^2 - 3x + 8) to yield 8x^4 - 12x^3 + 32x. For factoring, you will work the same problem backwards. For instance, you could start with the polynomial 8x^2 - 12x^3 + 32x and work backwards to 4x(2x - 3x^2 + 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.

\begin{array}{rrl} 15&=&3\times 5 \\ 24&=&2\times 2\times 2\times 3\text{ or }2^3\times 3 \\ 27&=&3\times 3\times 3\text{ or }3^3 \end{array}

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 24x^4y^2z, 18x^2y^4, and 12x^3yz^5.

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

\begin{array}{lll} 24x^4y^2z&=&2^3\cdot 3\cdot x^4\cdot y^2\cdot z \\ 18x^2y^4&=&2\cdot 3^2\cdot x^2\cdot y^4 \\ 12x^3yz^5&=&2^2\cdot 3\cdot x^3\cdot y\cdot z^5 \end{array}

By observation, what is shared between all three monomials is 2\cdot 3\cdot x^2\cdot y or 6x^2y.

Questions

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

  1. 9+8b^2
  2. x-5
  3. 45x^2 - 25
  4. 1 + 2n^2
  5. 56 - 35p
  6. 50x - 80y
  7. 7ab - 35a^2b
  8. 27x^2y^5 - 72x^3y^2
  9. -3a^2b + 6a^3b^2
  10. 8x^3y^2 + 4x^3
  11. -5x^2 - 5x^3 - 15x^4
  12. -32n^9+32n^6+40n^5
  13. 28m^4+40m^3+8
  14. -10x^4+20x^2+12x
  15. 30b^9+5ab-15a^2
  16. 27y^7+12y^2x+9y^2
  17. -48a^2b^2-56a^3b-56a^5b
  18. 30m^6+15mn^2-25
  19. 20x^8y^2z^2+15x^5y^2z+35x^3y^3z
  20. 3p+12q-15q^2r^2
  21. -18n^5+3n^3-21n+3
  22. 30a^8+6a^5+27a^3+21a^2
  23. -40x^{11}-20x^{12}+50x^{13}-50x^{14}
  24. -24x^6-4x^4+12x^3+4x^2
  25. -32mn^8+4m^6n+12mn^4+16mn
  26. -10y^7+6y^{10}-4y^{10}x-8y^8x

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-7-1/”>Answer Key 7.1

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