6.4 General Strategy for Factoring Polynomials

Topics covered in this section are:

Recognize and use the appropriate method to factor a polynomial completely

6.4.1 Recognize and Use the Appropriate Method to Factor a Polynomial Completely

GENERAL STRATEGY FOR FACTORING POLYNOMIALS

HOW TO: Use a general strategy for factoring polynomials.

Is there a greatest common factor?
Factor it out.
Is the polynomial a binomial, trinomial, or are there more than three terms?
If it is a binomial:
- Is it a sum?
  Of squares? Sums of squares do not factor.
  Of cubes? Use the sum of cubes pattern.
- Is it a difference?
  Of squares? Factor as the product of conjugates.
  Of cubes? Use the difference of cubes pattern.If it is a trinomial:
- Is it of the form $x^{2}+bx+c$? Undo FOIL.
- Is it of the form $ax^{2}+bx+c$?
  If $a$ and $c$ are squares, check if it fits the trinomial square pattern.
  Use the trial and error or “ac” method.
  If it has more than three terms:
- Use the grouping method.
Check.
Is it factored completely?
Do the factors multiply back to the original polynomial?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Example 1

Factor completely: $7x^{3}-21x^{2}-70x$.

Solution

	$7x^{3}-21x^{2}-70x$
Is there a GCF? Yes, $7x$.
Factor out the GCF.	$7x(x^{2}-3x-10)$
In the parentheses, is it a binomial, trinomial, or are there more terms?
It is a trinomial with leading coefficient $1$.
“Undo” FOIL.	$7x(x \ \ \ \ \ )(x \ \ \ \ \ \ )$
	$7x(x+2)(x-5)$
Is the expressions completely factored? Yes, neither binomial can be factored.
Check your answer by multiplying.	$7x(x+2)(x-5)$ $7x(x^{2}-5x+2x-10)$ $7x(x^{2}-3x-10$ $7x^{3}-21x^{2}-70x \ \checkmark$

Be careful when you are asked to factor a binomial as there are several options!

Example 2

Factor completely: $24y^{2}-150$.

Solution

	$24y^{2}-150$
Is there a GCF? Yes, $6$.
Factor out the GCF.	$6(4y^{2}-25)$
In the parentheses, is it a binomial, trinomial, or are there more than three terms. Binomial.
Is it a sum? No.
Is it a difference? Of squares or cubes? Yes, difference of squares.	$6((2y)^{2}-(5)^{2})$
Write as a product of conjugates.	$6(2y-5)(2y+5)$
Is the expression factored completely? Yes, neither binomial can be factored.
Check:	$6(2y-5)(2y+5)$ $6(4y^{2}-10y+10y-25)$ $24y^{2}-150 \ \checkmark$

The next example can be factored using several methods. recognizing the trinomial squares pattern will make your work easier.

Example 3

Factor completely: $4a^{2}-12ab+9b^{2}$.

Solution

	$4a^{2}-12ab+9b^{2}$
Is there a GCF? No.
Is it a binomial, trinomial, or are there more terms?
Trinomial with $a≠1$. But the first term is a perfect square.
Is the last term a perfect square? Yes.	$(2a)^{2}-12ab+(3b)^{2}$
Does it fit the pattern, $a^{2}-2ab+b^{2}$? Yes.	$(2a)^{2} \searrow – 12ab + \swarrow (3b)^{2}$ $-2(2a)(3b)$
Write it as a square.	$(2a-3b)^{2}$
Is the expression factored completely? Yes, the binomial cannot be factored.
Check your answer by multiplying.	$(2a-3b)^{2}$ $(2a)^{2}-2 \cdot 2a \cdot 3b + (3b)^{2}$ $4a^{2}-12ab+9b^{2} \ \checkmark$

Remember, sums of squares do not factor, but sums of cubes do!

Example 4

Factor completely: $12x^{3}y^{2}+75xy^{2}$.

Solution

	$12x^{3}y^{2}+75xy^{2}$
Is there a GCF? Yes, $3xy^{2}$.
Factor out the GCF.	$3xy^{2}(4x^{2}+25)$
In the parentheses, is it a binomial, trinomial, or are there more than three terms? It is a binomial.
Is it a sum? Of squares? Yes.	Sums of squares are prime.
Is the expression factored completely? Yes.
Check:	$3xy^{2}(4x^{2}+25)$ $12x^{3}y^{2}+75xy^{2} \ \checkmark$

When using the sum or difference of cubes pattern, be careful with the signs.

Example 5

Factor completely: $24x^{3}+81y^{3}$.

Solution

	$24x^{3}+81y^{3}$
Is there a GCF? Yes, $3$. Factor it out.	$3(8x^{3}+27y^{3})$
In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial.
Is it a sum or difference? Sum.
Sum of squares or cubes? Cubes.
Write it using the sum of cubes pattern.
Is the expression factored completely? Yes.	$3(2x+3y)(4x^{2}-6xy+9y^{2})$
Check by multiplying.	We leave this to you.

Example 6

Factor completely: $3x^{5}y-48xy$.

Solution

	$3x^{5}y-48xy$
Is there a GCF? Yes, $3xy$. Factor it out.	$3xy(x^{4}-16)$
Is the binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.	$3xy \left( (x^{2})^{2}-(4)^{2} \right)$
Factor it as a product of conjugates.	$3xy(x^{2}-4)(x^{2}+4)$
The first binomial is again a difference of squares.	$3xy \left( (x)^{2}-(2)^{2} \right) (x^{2}+4)$
Factor it as a product of conjugates.	$3xy(x-2)(x+2)(x^{2}+4)$
Is the expression factored completely? Yes.
Check by multiplying.	$3xy(x-2)(x+2)(x^{2}+4)$ $3xy(x^{2}-4)(x^{2}+4)$ $3xy(x^{4}-16)$ $3x^{5}y-48xy \ \checkmark$

Example 7

Factor completely: $4x^{2}+8bx-4ax-8ab$.

Solution

	$4x^{2}+8bx-4ax-8ab$
Is there a GCF? Factor out the GCF, $4$.	$4(x^{2}+2bx-ax-2ab)$
There are four terms, use grouping.	$4[x(x+2b)-a(x+ab)]$ $4(x+2b)(x-a)$
Is the expression completely factored? Yes.
Check your answer by multiplying.	$4(x+2b)(x-a)$ $4(x^{2}-ax+2bx-2ab)$ $4x^{2}+8bx-4ax-8ab \ \checkmark$

Taking out the complete GCF in the first step will always make your work easier.

Example 8

Factor completely: $40x^{2}+44xy-24y$.

Solution

	$40x^{2}y+44xy-24y$
Is there a GCF? Factor out the GCF, $4y$.	$4y(10x^{2}+11x-6)$
Factor the trinomial with $a≠1$.	$4y(5x-2)(2x+3)$
Is the expression completely factored? Yes.
Check your answer by multiplying.	$4y(5x-2)(2x+3)$ $4y(10x^{2}+11x-6)$ $40x^{2}y+44xy-24y \ \checkmark$

When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.

Example 9

Factor completely: $9x^{2}-12xy+4y^{2}-49$.

Solution

	$9x^{2}-12xy+4y^{2}-49$
Is there a GCF? No.
With more than $3$ terms, use grouping. Last $2$ terms have no GCF. Try grouping first $3$ terms.
Factor the trinomial with $a≠1$. But the first term is a perfect square.	$(3x)^{2}-12xy+(2y)^{2}-49$
Is the last term of the trinomial a perfect square? Yes.
Does the trinomial fit the pattern, $a^{2}-2ab+b^{2}$? Yes.	$(3x)^{2} _\searrow -12xy+ _\swarrow (2y)^{2}$ $-2(3x)(2y)$
Write the trinomial as a square.	$(3x-2y)^{2}-49$
Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.	$(3x-2y)^{2}-7^{2}$
Write it as a product of conjugates.	$((3x-2y)-7)((3x-2y)+7)$ $(3x-2y-7)(3x-2y+7)$
Is the expression factored completely? Yes.