Read: Special Cases - Squares
Learning Objectives
- Factor a polynomial of the form:
- Factor a polynomial of the form:
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Factoring a Perfect Square Trinomial
A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.A General Note: Perfect Square Trinomials
A perfect square trinomial can be written as the square of a binomial:Exercises
Factor .Answer: First, notice that and are perfect squares because and . This means that Next, check to see if the middle term is equal to , which it is:
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Therefore, the trinomial is a perfect square trinomial and can be written as .Answer
Example
Factor .Answer: First, notice that and are perfect squares because and . This means that , we could say that , but would that give a middle term of ? We will need to choose to get the results we want:
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Therefore, the trinomial is a perfect square trinomial and can be written as .Answer
Given a perfect square trinomial, factor it into the square of a binomial.
- Confirm that the first and last term are perfect squares.
- Confirm that the middle term is twice the product of .
- Write the factored form as , or.
Factoring a Difference of Squares
A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.A General Note: Differences of Squares
A difference of squares can be rewritten as two factors containing the same terms but opposite signs.Example
Factor .Answer: Notice that and are perfect squares because and . This means that The polynomial represents a difference of squares and can be rewritten as . Check that you are correct by multiplying.
Answer
Example
Factor .Answer: Notice that and are perfect squares because and . This means that The polynomial represents a difference of squares and can be rewritten as . Check that you are correct by multiplying.
Answer
How To: Given a difference of squares, factor it into binomials.
- Confirm that the first and last term are both perfect squares. For exponents, this means the exponent must be divisible by 2.
- Write the factored form as .
Think About It
Is there a formula to factor the sum of squares, , into a product of two binomials? Write down some ideas for how you would answer this in the box below before you look at the answer. [practice-area rows="1"][/practice-area]Answer: There is no way to factor a sum of squares into a product of two binomials, this is because of addition - the middle term needs to "disappear" and the only way to do that is with opposite signs. to get a positive result, you must multiply two numbers with the same signs. The only time a sum of squares can be factored is if they share any common factors, as in the following case: The only way to factor this expression is by pulling out the GCF which is 9.