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Algebra 1

6.7.2 Reviewing General Strategies to Factor a Polynomial

Algebra 16.7.2 Reviewing General Strategies to Factor a Polynomial

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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.

A chart titled General Strategy for Factoring Polynomials lists factoring methods: GCF; Binomial (difference and sum of squares); Trinomial (factoring methods); and polynomials with more than 3 terms (grouping).

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
4x2+20xy+25y24x2+20xy+25y2 After GCF, trinomial of form a2+2ab+b2a2+2ab+b2 trinomial square pattern

1. 8y3+16y224y8y3+16y224y

2. 16x336x16x336x

3. 6pq29pq6p6pq29pq6p

4. 4x212xy+9y2254x212xy+9y225

5. a2+9a2+9

6. 9x224xy+16y29x224xy+16y2

7. 27y24827y248

Are you ready for more?

Extending Your Thinking

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

9x212xy+4y2499x212xy+4y249

Self Check

Which strategy would you use to factor 40 x 2 y + 44 x y 24 y ? What is the factored result?
  1. Factor out the GCF. Use the perfect square trinomial strategy.

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

  2. Factor out the GCF. Use the " a c " method.

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

  3. Factor out the GCF. Use the difference of squares.

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

  4. Factor out the GCF. Use the undo FOIL method.

    4 y ( 5 x + 2 ) ( 2 x 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.

A chart titled General Strategy for Factoring Polynomials lists factoring methods: GCF; Binomial (difference and sum of squares); Trinomial (factoring methods); and polynomials with more than 3 terms (grouping).

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 x2+bx+cx2+bx+c? Undo FOIL.
  • Is it of the form ax2+bx+cax2+bx+c?
    • If aa and cc are squares, check if it fits the trinomial square pattern.
    • Use trial and error or the “acac” 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: 24y215024y2150.

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

Step 1 - Is there a GCF?

24y215024y2150

Yes, 66.

Step 2 - Factor out the GCF.

6(4y225)6(4y225)

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((2y)2(5)2)6((2y)2(5)2)

Step 6 - Write as a product of conjugates.

6(2y5)(2y+5)6(2y5)(2y+5)

Step 7 - Is the expression factored completely?

Neither binomial can be factored.

Step 8 - Check your answer by multiplying.

6(2y5)(2y+5)6(2y5)(2y+5)

6(4y225)6(4y225)

24y215024y2150

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
x23x10x23x10
4a212ab+9b24a212ab+9b2
4y2254y225

Try it

Try It: Reviewing General Strategies to Factor a Polynomial

Identify the strategy you would use to factor each polynomial.

1. 4x2y16xy+12y4x2y16xy+12y

2. 4x2+8bx4ax8ab4x2+8bx4ax8ab

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