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Simple Trinomials as Products of Binomials
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Multiplying Fractions
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Factoring Trinomials
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Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
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Factoring a Polynomial by Finding the GCF
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Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
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Finding the GCF of a Set of Monomials
 
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Factoring Trinomials

Sometimes we must use several methods to completely factor a polynomial.

Procedure — General Strategy for Factoring a Polynomial

Step 1 Factor out the GCF of the terms of the polynomial.

Step 2 Count the number of terms and look for factoring patterns.

Two terms: Look for the difference of two squares, the sum of two cubes, or the difference of two cubes.

Three terms:

• Try to factor using the patterns for a perfect square trinomial.

• If the trinomial has the form x2 + bx + c, try to find two integers whose product is c and whose sum is b.

• If the trinomial has the form ax2 + bx + c, try to find two integers whose product is ac and whose sum is b.

Four terms: Try factoring by grouping.

Step 3 Factor completely.

To check the factorization, multiply the factors.

 

Example 1

Factor: 108wx2 - 36wxy + 3wy2

Solution

Step 1 Factor out the GCF.

Factor each term.

The GCF is 3w.

Factor out 3w.

108wx2

= 2 · 2 · 3 · 3 · 3 · w · x · x

= 2 · 2 · 3 · 3 · 3 · w · x · x

= 3w(36x2 - 12xy + y2)

- 36wxy

- 2 · 2 · 3 · 3 · w · x · y

- 2 · 2 · 3 · 3 · w · x · y

 

+ 3wy2

+ 3 · w · y · y

+ 3 · w · y · y

Step 2 Count the number of terms and look for factoring patterns.

There are three terms in 36x2 - 12xy + y2.

The first term, 36x2, is a perfect square, (6x)2.

The third term, y2, is a perfect square, (y)2.

The middle term, 12xy, is twice the product of the squared terms: 12xy = 2(6x)(y)

This matches the pattern for a perfect square trinomial.

Substitute 6x for a and y for b: a2  2ab  b2 `(a  b)2

36x2 - 12xy + y2 = (6x)2 - 2(6x)(y) + (y)2 = (6x - y)2

Step 3 Factor completely.

(6x - y)2 cannot be factored further.

Thus, the factorization is 3w(6x - y)2.

You can multiply to check the factorization.

Note:

Don’t forget the GCF, 3w, that we factored out in Step 1.

 

Some polynomials cannot be factored using integers.

 

Example 2

Factor: x2 + 3x + 5

Solution

Step 1 Factor out the GCF.

There are no factors common to all three terms, other than 1 or 1.

Step 2 Count the number of terms and look for factoring patterns.

There are three terms in x2 + 3x + 5.

This trinomial has the form ax2 + bx + c.

To factor this trinomial we must find two integers whose product is 5 and whose sum is 3. Here are the possibilities:

Product

1 · 5

-1 · -5

Sum

6

-6

There are no integers whose product is 1 and whose sum is 3.

So, the trinomial x2 + 3x + 5 cannot be factored using integers.

 

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