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Factoring – Review of Basic Factoring

• The following is a quick review of basic factoring

1. Common Factor

3 25 10 20x x x

Example 1:

25 2 4x x x

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2. Difference of two squares

2 2a b

Pattern:

2 16x

a b a b

Example 2:

4 4x x

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3. Sum of two squares

2 2a bPattern:

A sum of two squares cannot be factored.

2 9x

Example 3:

Try to factor the following:

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3 3x x

We can’t find two factors that multiply to give us the sum of two squares,

3 3x x

3 3x x

2 9x

2 6 9x x

2 6 9x x

2 9x

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4. Difference of two cubes

3 3a b

Pattern:

3 8x

2 2a b a ab b

Example 4:

2( 2)( 2 4)x x x

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5. Sum of two cubes

3 3a b

Pattern:

3 27x

2 2a b a ab b

Example 5:

2( 3)( 3 9)x x x

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6. Perfect square trinomial

2 22a ab b

Pattern:

2 22a ab b

2a b

2a b

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2 12 36x x

( )x

Example 6: Perfect square trinomial

6 6

2 6 12

x x

x x

( 6)x

What do you square to get the first term of x squared?

What do you square to get the last term of 36?

Find the product of these two terms and then double it.

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2 12 36x x

Since the result of 12x is the same as the middle term …

… the trinomial is a perfect square trinomial.

Take the binomial … ( 6)x

and insert a plus sign (same sign as the middle term)

( 6)x

Square the binomial and the trinomial is factored.

2( 6)x

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2 6 9x x

( )x

Example 7: Perfect square trinomial

3 3

2 3 6

x x

x x

( 3)x

What do you square to get the first term of x squared?

What do you square to get the last term of 9?

Find the product of these two terms and then double it.

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2 6 9x x

Since the result of 6x is the same as the middle term …

… the trinomial is a perfect square trinomial.

Take the binomial … ( 3)x

and insert a negative sign (same sign as the middle term)

( 3)x

Square the binomial and the trinomial is factored.

2( 3)x

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7. Trinomial

There are two methods for factoring a trinomial, guess and check, and the ac method.

The guess and check method will be reviewed here. See the authors website for help with the ac method.

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2 6x x

( )( )

Example 8: Trinomial

Write two binomials.

The product of the first terms of the binomials must equal the first term of the trinomial.

Since the third term of the trinomial is negative, the signs must be opposite.

( )( )x x

( )( )x x

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2 6x x

Consider the possible factors of the third term, 6.

( )( )x x

1 6 6

2 3 6

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2 6x x

Try the different pairs of factors in the binomial, and see if the outside and inside matches the middle term.

1 6 6

2 3 6

( 1)( 6)x x

6 1 5x x x 3 2x x x

( 2)( 3)x x

No Yes

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2 6x x

The correct factorization of the trinomial is …

( 2)( 3)x x

Remember that if the outside and inside yields the right numerical value, but opposite in sign, simply switch the two signs and the trinomial is factored.

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26 11 3x x

( )( )

Example 9: Trinomial

Write two binomials.

The product of the first terms of the binomials must equal the first term of the trinomial.

Since the value of a is not 1 in this case, we must consider the possible factorizations of 6.

1a

1 6 6

2 3 6

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26 11 3x x

Try 1 and 6 along with the values of x to get the first term.

(1 )(6 )x x

Since the third term of the trinomial is positive, the signs must be the same.

Since the middle term of the trinomial is negative, the signs are both negative.

( )(6 )x x

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26 11 3x x

Consider the possible factorizations of the third term.

1 3 3

We now have a number of possibilities to try. For the leading term we have …

1 6 6

2 3 6

and for the last term we have …

1 3 3

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Try the first combination, and see if the outside and inside matches the middle term.

( 1)(6 3)x x

3 6 9x x x

No

26 11 3x x

1 6 6

2 3 6

1 3 3

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Since the leading coefficient was not 1, we need to turn around the last term’s values and try again.

( 3)(6 1)x x

18 19x x x

No

26 11 3x x

1 6 6

2 3 6

1 3 3

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Note that we did get a different value than the first time, but it still does not match the middle term of the trinomial.

18 19x x x

No

26 11 3x x

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Try the other pair of factors of the first term, with the same pair of factors for the last term.

(2 1)(3 3)x x

6 3 9x x x

No

26 11 3x x

1 6 6

2 3 6

1 3 3

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Once again, turn around the last pair.

(2 3)(3 1)x x

2 9 11x x x

Yes

26 11 3x x

1 6 6

2 3 6

1 3 3

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The factorization of the trinomial is given by …

(2 3)(3 1)x x

26 11 3x x

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8. Grouping method

Consider the grouping method when there are four terms.

Example 10:

3 25 7 35x x x

Group the polynomial into two binomials.

3 25 7 35x x x

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Factor each binomial.

3 25 7 35x x x

2 5 7 5x x x

Factor out the common binomial factor

2( 5) 7x x

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