Factoring Strategy

Section 6.6 Factoring Strategy

Objective: Idenfity and use the correct method to factor various polynomials.

With so many different tools used to factor, it is easy to get lost as to which tool to use when. Here we will attempt to organize all the different factoring types we have seen. A large part of deciding how to solve a problem is based on how many terms are in the problem. For all problem types we will always try to factor out the GCF first.

\begin{equation*} \textbf{ Factoring Strategy (GCF First!!!!!) } \end{equation*}

2 terms: sum or difference of squares or cubes:
\begin{align*} a^2 - b^2 = \amp (a + b)(a - b) \\ a^3 +b^3 = \amp (a+b)(a^2 - ab+b^2)\\ a^3 - b^3 = \amp (a - b)(a^2 + ab + b^2) \end{align*}
3 terms: ac method, watch for perfect square!
\begin{gather*} a^2 +2ab+b^2 =(a+b)^2\\ \text{Multiply to } ac \text{ and add to } b \end{gather*}
4 terms: grouping

We will use the above strategy to factor each of the following examples. Here the emphasis will be on which strategy to use rather than the steps used in that method.

Example 6.6.1.

\begin{align*} 4x^2 + 56xy + 196y^2 \amp \quad \text{ GCF first, } 4 \\ 4(x^2 + 14xy + 49y^2) \amp\quad \text{ Three terms, try $ac$ method, multiply to $49$, add to } 14 \\ \amp\quad \text{$7$ and $7$, perfect square! } \\ 4(x + 7y)^2 \amp\quad \text{ Our Solution } \end{align*}

Example 6.6.2.

\begin{align*} 5x^2y + 15xy - 35x^2 - 105x \amp \quad \text{ GCF first, } 5x \\ 5x(xy + 3y - 7x - 21) \amp\quad \text{ Four terms, try grouping } \\ 5x[y(x+3)-7(x+3)] \amp\quad (x+3) \text{ match! } \\ 5x(x + 3)(y - 7) \amp\quad \text{ Our Solution } \end{align*}

Example 6.6.3.

\begin{align*} 100x^2 - 400 \amp \quad \text{ GCF first, } 100 \\ 100(x^2 - 4) \amp\quad \text{ Two terms, difference of squares} \\ 100(x + 4)(x - 4) \amp\quad \text{ Our Solution } \end{align*}

Example 6.6.4.

\begin{align*} 108x^3y^2 - 39x^2y^2 + 3xy^2 \amp \quad \text{ GCF first, } 3xy^2 \\ 3xy^2(36x^2 - 13x + 1) \amp\quad \text{ Thee terms, $ac$ method, multiply to $36$, add to } -13 \\ 3xy^2(36x^2-9x-4x+1) \amp\quad -9 \text{ and $-4$,split middle term } \\ 3xy^2[9x(4x - 1) - 1(4x - 1)] \amp\quad \text{ Factor by grouping } \\ 3xy^2(4x - 1)(9x - 1) \amp\quad \text{ Our Solution } \end{align*}

World View Note:Variables originated in ancient Greece where Aristotle would use a single capital letter to represent a number.

Example 6.6.5.

\begin{align*} 5 + 625y^3 \amp \quad \text{ GCF first, } 5 \\ 5(1 + 125y^3) \amp\quad \text{ Two terms, sum of cubes } \\ 5(1 + 5y)(1 - 5y + 25y^2) \amp\quad \text{ Our Solution } \end{align*}

It is important to be comfortable and confident not just with using all the fac- toring methods, but decided on which method to use. This is why practice is very important!

Exercises Exercises - Factoring Strategy

Exercise Group.

Factor each completely.

1.

$24az -18ah+60yz -45yh$

2.

$2x^2 -11x+15$

3.

$5u^2-9uv+4v^2 $

4.

$16x^2+48xy+36y^2$

5.

$-2x^3+128y^3$

6.

$20uv-60u^3-5xv+15xu^2$

7.

$5n^3+7n^2-6n $

8.

$2x^3+5x^2y+3y^2x$

9.

$54u^3 - 16 $

10.

$54 - 128x^3 $

11.

$n^2-n $

12.

$5x^2-22x-15 $

13.

$x^2-4xy+3y^2$

14.

$45u^2-150uv+125v^2$

15.

$9x^2 - 25y^2 $

16.

$x^3 - 27y^3 $

17.

$m^2 -4n^2$

18.

$12ab-18a+6nb-9n$

19.

$36b^2c - 16xd - 24b^2d + 24xc $

20.

$3m^3 - 6m^2n - 24n^2m$

21.

$128 + 54x^3$

22.

$64m^3 + 27n^3$

23.

$2x^3 + 6x^2y - 20y^2x$

24.

$3ac + 15ad^2 + x^2c + 5x^2d^2$

25.

$n^3 +7n^2 +10n$

26.

$64m^3 -n^3 $

27.

$27x^3 - 64 $

28.

$16a^2 - 9b^2$

29.

$5x^2 +2x$

30.

$2x^2 -10x+12 $

31.

$3k^3 - 27k^2 + 60k $

32.

$32x^2 - 18y^2$

33.

$mn-12x+3m-4xn $

34.

$2k^2+k-10 $

35.

$16x^2-8xy+y^2 $

36.

$v^2+v$

37.

$27m^2 - 48n^2$

38.

$x^3 + 4x^2$

39.

$9x^3 + 21x^2y - 60y^2x$

40.

$9n^3 - 3n^2$

41.

$2m^2 + 6mn - 20n^2$

42.

$2u^2v^2 - 11uv^3 + 15v^4$