Factoring Perfect Square Trinomials and the Difference of Squares

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Factoring Perfect Square Trinomials and the Difference of Squares

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Factoring Perfect Square Trinomials and the Difference of Squares

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Solve

Notice that if you take ½ of the middle number and square it, you get the last number. 6 divided by 2 is 3, and 32 is 9. When this happens you have a special product.

This problem factors into

(a + b)² = (a + b)(a + b) = a² +ab + ab+ b² = a² +2ab +b²

(a - b)² = (a - b)(a - b) = a² -ab - ab+ b² = a² -2ab +b²

a²

2ab

b²

For example: (3x + 5)² = (3x)² + 2(3x)(5) + (5)² = 9x² + 30x +25

To factor this trinomial , the grouping method can be used. However, if we recognize that the trinomial is a perfect square trinomial, we can use one of the following patterns to reach a quick solution

a = 3x b = 5

- Check if the first and third terms are both perfect squares with positive coefficients.
- If this is the case, identify a and b, and determine if the middle term equals 2ab

t² + 10t +25

- The first term is a perfect square: t² = (t)²
- The third term is a perfect square: 25 = (5)²
- The middle term is twice the product of t and 5: 2(t)(5)

t² + 10t +25 Perfect square trinomial

- Check if the first and third terms are both perfect squares with positive coefficients.
- If this is the case, identify a and b, and determine if the middle term equals 2ab

t² + 4t +1

- The first term is a perfect square: t² = (t)²
- The third term is a perfect square: 1 = (1)²
- The middle term is not twice the product of t and 1: 2(t)(1)

t² + 4t + 1Is not perfect square trinomial

Double the product of x and 6

Remember: A perfect square trinomial is one that can be factored into two factors that match each other (and hence can be written as the factor squared).

This is a perfect square trinomial because it factors into two factors that are the same and the middle term is twice the product of x and 6. It can be written as the factor squared.

Notice that the first and last terms are perfect squares.

The middle term comes from the outers and inners when FOILing. Since they match, it ends up double the product of the first and last term of the factor.

The GCF is 1.

The first and third terms are positive

The first term is a perfect square: 25y² = (5y)²

The third term is a perfect square: 4 = (2)²

The middle term is twice the product of 5y and 2: 20y = 2(5y)(2)

Factor as (5y - 2)²

y² - 25

The binomial is a difference of squares.

= (y)² - (5)²

Write in the form: a² - b², where a = y, b = 5.

Factor as (a + b)(a – b)

= (y + 5)(y – 5)

difference

When you see two terms, look for the difference of squares. Is the first term something squared? Is the second term something squared but with a minus sign (the difference)?

{

rhyme for the day

The difference of squares factors into conjugate pairs!

A conjugate pair is a set of factors that look the same but one has a + and one has a – between the terms.

Factor Completely:

Look for something in common (there is a 5)

Two terms left----is it the difference of squares?

Yes---so factor into conjugate pairs.

Look for something in common

There is a 2p in each term

Three terms left---try trinomial factoring

"unFOILing"

Check by FOILing and then distributing 2p through

Factoring a Sum and Difference of Cubes

Sum of Cubes: a³+ b³ = (a + b)(a² -ab+b²)

Difference of Cubes: a³ - b³ = (a - b)(a² +ab+b²)

Square the first term of the binomial

x³ + 8 = (x)³ + (2) ³

-(x)(2)

= (x + 2) ( (x)² - (x)(2) + (2)²)

(x + 2)

x

2

Product of terms in the binomial

Square the last term of the binomial.

x³ and 8 are perfect cubes

- The factored form is the product of a binomial and a trinomial.
- The first and third terms in the trinomial are the squares of the terms within the binomial factor.
- Without regard to signs, the middle term in the trinomial is the product of terms in the binomial factor.

multiply together but change sign

square the first term

square the last term

can be sum or difference here

If it's not the difference of squares, see if it is sum or difference of cubes. Is the first term something cubed (to the third power)? Is the second term something cubed?

The first factor comes from what was cubed.

You must just memorize the steps to factor cubes. You should try multiplying them out again to assure yourself that it works.

Let's try one more:

multiply together but change sign

square the first term

square the last term

What cubed gives the first term?

What cubed gives the second term?

The first factor comes from what was cubed.

Try to memorize the steps to get the second factor:

First term squared---multiply together & change sign---last term squared