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Algebra 1

6.7.3 Implementing General Strategies for Factoring

Algebra 16.7.3 Implementing General Strategies for Factoring
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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 5 y 3 − 15 y 2 − 270 y

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

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2. 75 m 3 + 12 m 75 m 3 + 12 m

3. 16 x 2 + 24 x y − 4 x − 6 y 16 x 2 + 24 x y − 4 x − 6 y

4. 2 n 2 + 13 n − 7 2 n 2 + 13 n − 7

5. 121 r 2 − s 2 121 r 2 − s 2

6. 25 w 2 − 60 w + 36 25 w 2 − 60 w + 36

7. ( 3 x + 1 ) 2 − 6 ( 3 x + 1 ) + 9 ( 3 x + 1 ) 2 − 6 ( 3 x + 1 ) + 9

8. 16 x 2 − 24 x y + 9 y 2 − 64 16 x 2 − 24 x y + 9 y 2 − 64

Video: Determining Factoring Method

Watch the following video to learn more about how to determine which method of factoring will be best to apply.

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Self Check

Which strategy would you use to factor 9 x 2 − 24 x y + 16 y 2 ? What is the factored result?
  1. trinomial square pattern; ( 3 x − 4 y ) 2
  2. difference of squares; ( 4 x − 3 y ) 2
  3. grouping; ( 3 x − 4 y ) ( 3 x + 4 y )
  4. trinomial square pattern; ( 3 x + 4 y ) 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 7 x 3 − 21 x 2 − 70 x .

Step 1 - Is there a GCF?

7 x 3 − 21 x 2 − 70 x 7 x 3 − 21 x 2 − 70 x

Yes, 7 x 7 x .

Step 2 - Factor out the GCF.

7 x ( x 2 − 3 x − 10 ) 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 1 .

Step 4 - “Undo” FOIL.

7 x ( x ) ( x ) 7 x ( x ) ( x )

7 x ( x + 2 ) ( x − 5 ) 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 ) ( x − 5 )

7 x ( x 2 − 5 x + 2 x − 10 ) 7 x ( x 2 − 5 x + 2 x − 10 )

7 x ( x 2 − 3 x − 10 ) 7 x ( x 2 − 3 x − 10 )

7 x 3 − 21 x 2 − 70 x ✓ 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 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 4 x 2 + 8 b x − 4 a x − 8 a b

Factor out the GCF, 4 4 .

4 ( x 2 + 2 b x − a x − 2 a b ) 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 ( x + 2 b ) − a ( x + 2 b ) ]

4 ( x + 2 b ) ( x − a ) 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 b ) ( x − a )

4 ( x 2 − a x + 2 b x − 2 a b ) 4 ( x 2 − a x + 2 b x − 2 a b )

4 x 2 + 8 b x − 4 a x − 8 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 40 x 2 y + 44 x y − 24 y .

Step 1 - Is there a GCF?

40 x 2 y + 44 x y − 24 y 40 x 2 y + 44 x y − 24 y

Factor out the GCF, 4 y 4 y .

4 y ( 10 x 2 + 11 x − 6 ) 4 y ( 10 x 2 + 11 x − 6 )

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

4 y ( 5 x − 2 ) ( 2 x + 3 ) 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 ( 5 x − 2 ) ( 2 x + 3 )

4 y ( 10 x 2 + 11 x − 6 ) 4 y ( 10 x 2 + 11 x − 6 )

40 x 2 y + 44 x y − 24 y ✓ 40 x 2 y + 44 x y − 24 y ✓

Try it

Try It: Implementing General Strategies for Factoring

Find a strategy that fits and factor completely:

1. 27 y 2 − 48 27 y 2 − 48

2. 6 x 2 − 12 x c + 6 b x − 12 b c 6 x 2 − 12 x c + 6 b x − 12 b c

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