Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

6.7.2 Reviewing General Strategies to Factor a Polynomial

6.7.2 • Reviewing General Strategies to Factor a Polynomial

Activity

Take a moment to look at all parts of the diagram. Are there any parts that you do not understand?

Use this time to ask your teacher if there is something you would like clarified here.

Use the diagram above as needed for the activity.

Determine the best strategy to factor the polynomial. Be prepared to show your reasoning. Remember to factor out the GCF first, if necessary, before choosing a strategy. If a polynomial is prime, write “prime.” An example is provided.

Polynomial	Reasoning	Strategy
$4 x^{2} + 20 x y + 25 y^{2}$	After GCF, trinomial of form $a^{2} + 2 a b + b^{2}$	trinomial square pattern

1. $8 y^{3} + 16 y^{2} - 24 y$

Compare your answer:

After GCF, trinomial of form $x^{2} + b x + c$ ; undo FOIL

2. $16 x^{3} - 36 x$

Compare your answer:

After GCF, a binomial of squares; difference of squares

3. $6 p q^{2} - 9 p q - 6 p$

Compare your answer:

After GCF, trinomial of form $a x^{2} + b x + c$ ; “ $a c$ ” method or trial and error

4. $4 x^{2} - 12 x y + 9 y^{2} - 25$

Compare your answer:

polynomial of 4 terms; grouping

5. $a^{2} + 9$

Compare your answer:

no factors; prime

6. $9 x^{2} - 24 x y + 16 y^{2}$

Compare your answer:

trinomial of form $a^{2} - 2 a b + b^{2}$ ; trinomial square pattern

7. $27 y^{2} - 48$

Compare your answer:

After GCF, a binomial of squares; difference of squares

Are you ready for more?

Extending Your Thinking

Factor the polynomial below. Write down every factoring strategy that you use.

$9 x^{2} - 12 x y + 4 y^{2} - 49$

Compare your answer:

$(3 x - 2 y - 7) (3 x - 2 y + 7)$ ; Strategies Used:

Factor out GCF, grouping, perfect square trinomial, difference of squares

Here is how to factor this polynomial:

Step 1 - Is there a GCF?

$9 x^{2} - 12 x y + 4 y^{2} - 49$

No.

Step 2 - With more than three terms, use grouping. The last two terms have no GCF. Try grouping the first three terms.

Step 3 - Factor the trinomial with $a \neq 1$ . But the first term is a perfect square.

Is the last term of the trinomial a perfect square?

$(3 x)^{2} - 12 x y + (2 y)^{2} - 49$

Yes.

Step 4 - Does the trinomial fit the pattern $a^{2} \pm 2 a b + b^{2}$ ?

${(3 x)}^{2}_{&searrow;} \underset{−2 (3 x) (2 y)}{−12 x y +}_{&swarrow;} {(2 y)}^{2} - 49$

Yes.

Step 5 - Write the trinomial as a square.

$(3 x - 2 y)^{2} - 49$

Step 6 - Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.

$(3 x - 2 y)^{2} - 7^{2}$

Step 7 - Write it as a product of conjugates.

$((3 x - 2 y) - 7) ((3 x - 2 y) + 7)$

$(3 x - 2 y - 7) (3 x - 2 y + 7)$

Step 8 - Is the expression factored completely?

Yes.

Step 9 - Check your answer by multiplying.

$(3 x - 2 y - 7) (3 x - 2 y + 7)$

$9 x^{2} - 6 x y - 21 x - 6 x y + 4 y^{2} + 14 y + 21 x - 14 y - 49$

$9 x^{2} - 12 x y + 4 y^{2} - 49 ✓$

Self Check

Which strategy would you use to factor

40x^2y + 44xy − 24y

? What is the factored result?

Factor out the GCF. Use the perfect square trinomial strategy.

$4y(5x + 2)(2x - 3)$
Factor out the GCF. Use the " $a c$ " method.

$4y(5x - 2)(2x + 3)$
Factor out the GCF. Use the difference of squares.

$4y(5x - 2)(5x + 2)$
Factor out the GCF. Use the undo FOIL method.

$4y(5x + 2)(2x - 3)$

Additional Resources

Reviewing General Strategies to Factor a Polynomial

You have now become acquainted with all the methods of factoring that you will need in this course. The following chart summarizes all the factoring methods we have covered and outlines a strategy you should use when factoring polynomials.

These are the steps to use when deciding how to factor a polynomial. We will review an example after looking at these steps.

Use a general strategy for factoring polynomials:

Step 1 - Is there a greatest common factor? Factor it out.

Step 2 - Is the polynomial a binomial, a 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.

If it is a trinomial:

Is it of the form $x^{2} + b x + c$ ? Undo FOIL.
Is it of the form ax2+bx+cax2+bx+c?
- If $a$ and $c$ are squares, check if it fits the trinomial square pattern.
- Use trial and error or the “ $a c$ ” method.

If it has more than three terms:

Use the grouping method.

Step 3 - Check.

Is it factored completely?
Do the factors multiply back to the original polynomial?

Example 1

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

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

Step 1 - Is there a GCF?

$24 y^{2} - 150$

Yes, $6$ .

Step 2 - Factor out the GCF.

$6 (4 y^{2} - 25)$

Step 3 - In the parentheses, is it a binomial, a trinomial, or are there more than three terms? Binomial.

Step 4 - Is it a sum?

No.

Step 5 - Is it a difference? Of squares or cubes?

Yes, squares.

$6 ((2 y)^{2} - (5)^{2})$

Step 6 - Write as a product of conjugates.

$6 (2 y - 5) (2 y + 5)$

Step 7 - Is the expression factored completely?

Neither binomial can be factored.

Step 8 - Check your answer by multiplying.

$6 (2 y - 5) (2 y + 5)$

$6 (4 y^{2} - 25)$

$24 y^{2} - 150 ✓$

Example 2

Use the diagram and questions as references. Which strategy would you use to factor the following polynomials in the left column of the table? Be prepared to show your reasoning.

Look at each polynomial. Review the steps to decide which strategy to use.

Polynomial	Strategy	Reasoning
$x^{2} - 3 x - 10$
$4 a^{2} - 12 a b + 9 b^{2}$
$4 y^{2} - 25$

Compare your answers:

Polynomial	Strategy	Reasoning
$x^{2} - 3 x - 10$	undo FOIL	trinomial of form $x^{2} + b x + c$
$4 a^{2} - 12 a b + 9 b^{2}$	trinomial square pattern	trinomial of form $a^{2} - 2 a b + b^{2}$
$4 y^{2} - 25$	difference of squares	a binomial of squares

Try it

Reviewing General Strategies to Factor a Polynomial

Identify the strategy you would use to factor each polynomial.

1. $4 x^{2} y - 16 x y + 12 y$

Compare your answer:

Often, there will be more than one strategy that can be used to factor a polynomial. Here is a sample solution.

Factor out the GCF of $4 y$ . Use the undo FOIL method on the remaining polynomial since it is of the form $x^{2} + b x + c$ .

2. $4 x^{2} + 8 b x - 4 a x - 8 a b$

Compare your answer:

Often, there will be more than one strategy that can be used to factor a polynomial. Here is a sample solution.

Factor out GCF of $4$ . Then factor by grouping since there are four terms.