Implementing General Strategies for Factoring

6.7.3 Implementing General Strategies for Factoring

6.7.3 • Implementing General Strategies for Factoring

Activity

Work with a partner on these problems. After each solution, discuss if there are multiple strategies that would work to factor the polynomial.

Find a strategy that fits and factor completely:

1. $5 y^{3} - 15 y^{2} - 270 y$

You can use the following video if you need some help.

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Compare your answer:

Factor out GCF of $5 y$ . Then use the “undo FOIL” method on the polynomial of the form $x^{2} + b x + c$ . $5 y (y - 9) (y + 6)$

2. $75 m^{3} + 12 m$

Compare your answer:

Factor out GCF of $3 m$ . $3 m (25 m^{2} + 4)$

3. $16 x^{2} + 24 x y - 4 x - 6 y$

Compare your answer:

Factor out GCF of $2$ . Then use the grouping method. $2 (4 x - 1) (2 x + 3 y)$

4. $2 n^{2} + 13 n - 7$

Compare your answer:

Use the “ $a c$ ” method or trial and error. $(2 n - 1) (n + 7)$

5. $121 r^{2} - s^{2}$

Compare your answer:

Use the difference of squares method. $(11 r - s) (11 r + s)$

6. $25 w^{2} - 60 w + 36$

Compare your answer:

Use the perfect square trinomial method. $(5 w - 6)^{2}$

7. $(3 x + 1)^{2} - 6 (3 x + 1) + 9$

Compare your answer:

Use the substitution or perfect square trinomial method. $(3 x - 2)^{2}$

8. $16 x^{2} - 24 x y + 9 y^{2} - 64$

Compare your answer:

Use the grouping method on the first three terms. They are a perfect square trinomial. Then use the difference of squares on the result. $(4 x - 3 y - 8) (4 x - 3 y + 8)$

Self Check

Which strategy would you use to factor

9x^2 − 24xy + 16y^2

? What is the factored result?

trinomial square pattern; $(3x - 4y)^2$
difference of squares; $(4x - 3y)^2$
grouping; $(3x - 4y)(3x + 4y)$
trinomial square pattern; $(3x + 4y)^2$

Additional Resources

Implementing General Strategies for Factoring

Let’s practice recognizing and implementing these strategies as we factor polynomials.

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

Example 1

Find a strategy that fits and factor completely: $7 x^{3} - 21 x^{2} - 70 x$ .

Step 1 - Is there a GCF?

$7 x^{3} - 21 x^{2} - 70 x$

Yes, $7 x$ .

Step 2 - Factor out the GCF.

$7 x (x^{2} - 3 x - 10)$

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

Trinomial with leading coefficient $1$ .

Step 4 - “Undo” FOIL.

$7 x (x) (x)$

$7 x (x + 2) (x - 5)$

Step 5 - Is the expression factored completely?

Yes. Neither binomial can be factored.

Step 6 - Check your answer by multiplying.

$7 x (x + 2) (x - 5)$

$7 x (x^{2} - 5 x + 2 x - 10)$

$7 x (x^{2} - 3 x - 10)$

$7 x^{3} - 21 x^{2} - 70 x ✓$

Example 2

Find a strategy that fits and factor completely: $4 x^{2} + 8 b x - 4 a x - 8 a b$ .

Step 1 - Is there a GCF?

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

Factor out the GCF, $4$ .

$4 (x^{2} + 2 b x - a x - 2 a b)$

Step 2 - There are four terms. Use grouping.

$4 [x (x + 2 b) - a (x + 2 b)]$

$4 (x + 2 b) (x - a)$

Step 3 - Is the expression factored completely?

Yes.

Step 4 - Check your answer by multiplying.

$4 (x + 2 b) (x - a)$

$4 (x^{2} - a x + 2 b x - 2 a b)$

$4 x^{2} + 8 b x - 4 a x - 8 a b ✓$

Example 3

Find a strategy that fits and factor completely: $40 x^{2} y + 44 x y - 24 y$ .

Step 1 - Is there a GCF?

$40 x^{2} y + 44 x y - 24 y$

Factor out the GCF, $4 y$ .

$4 y (10 x^{2} + 11 x - 6)$

Step 2 - Factor the trinomial with $a \neq 1$ using the “ $a c$ ” method or trial and error.

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

Step 3 - Is the expression factored completely?

Yes.

Step 4 - Check your answer by multiplying.

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

$4 y (10 x^{2} + 11 x - 6)$

$40 x^{2} y + 44 x y - 24 y ✓$

Try it

Implementing General Strategies for Factoring

Find a strategy that fits and factor completely:

1. $27 y^{2} - 48$

2. $6 x^{2} - 12 x c + 6 b x - 12 b c$

Here is how to factor these polynomials:

1. difference of squares; $3 (3 y - 4) (3 y + 4)$

2. grouping; $6 (x + b) (x - 2 c)$