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Factoring – Trinomials (a = 1)

2ax bx c If leading coefficient a =1, we have …

• We will start factoring trinomials where a = 1, that is, the leading coefficient is 1. A trinomial in variable x is given below:

21x bx c 2x bx c

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( 3)( 5)x x

• Remember that a trinomial is often the result of multiplying two binomials.

• Example 1

2 2 15x x

Trinomial

FOIL

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• Our goal is to turn this process around. Start with the trinomial, and produce the product of binomials. To do this, use the steps of FOIL.

First

OutsideInside

Last

• Important: one of the steps in the factoring process is to determine the signs. If you have not already studied the previous slideshow on signs, you should do so now.

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2 6x x

( )( )

Example 2

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 binomials, and see if the outside/inside matches the middle term.

1 6 6

2 3 6

( 1)( 6)x x

6x 3x

( 2)( 3)x x

No Yes

1x 5x 2x x

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2 6x x

The correct factorization of the trinomial is …

( 2)( 3)x x

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2 6 16x x

( )( )

Example 3

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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Consider the possible factors of the third term, 16.

( )( )x x

1 16 16

2 8 16

4 4 16

2 6 16x x

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2 6 16x x

( 1)( 16)x x

Binomials Outside + Inside

16 1 15x x x No

( 2)( 8)x x 8 2 6x x x No

( 4)( 4)x x 4 4 0x x No

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

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2 6 16x x

Note that the second pair was very close to giving the correct value of the middle term.

( 2)( 8)x x 8 2 6x x x

The following rule is important to remember in this special case.

The result is the same, except for the sign.

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If the outside/inside yields the right numerical value of the middle term, but opposite in sign, simply switch the two signs and the trinomial is factored.

( 2)( 8)x x

8 2 6x x x Switch the signs. ( 2)( 8)x x Determine the outside and inside.

This is now the correct middle term, and the trinomial is factored.

( 2)( 8)x x

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One last comment on this problem. We went into great detail to make sure the process was understood. Now, lets simplify and use the quick method.

This works when both first terms of the binomials have coefficients of 1.

Recall the possible last terms …

2 6 16x x ( )( )x x

1 16 16

2 8 16

4 4 16

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Since the signs of the binomials are opposite, determine which pair of numbers has a difference that matches the numerical value (ignoring sign) of the middle term.

Pairs

1 16

2 8

4 4

Difference

15

6

0

The second pair of 2 and 8 is the one we want.

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Put the 2 and the 8 into the binomials, and if the outside plus inside gives the wrong sign, just switch signs.

In a problem where both signs of the binomials are the same, find the sum of the pairs of numbers to see which pair gives the correct middle term.

2 6 16x x ( 2)( 8)x x

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SUMMARY

To factor a trinomial of the form2x bx c

1. Write the binomials with first terms

2. Determine the signs.

3. Determine the possible factors of the third term.

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4. Find which pair of factors as last terms in the binomials will yield an outside/inside term equal to the middle term.

If the signs of the binomials are: a) opposite – take the difference of the

pairs of factors b) same – take the sum of the pairs of

factors

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2 7 12x x

Example 4

( )( )x x

( )( )x x

1. Write the binomials with first terms

2. Determine the signs.

Third term positive – signs are the sameMiddle term negative – both are negative

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2 7 12x x

1 12

2 6

3 4

3. Determine the possible factors of the third term.

4. Find which pair of factors as last terms in the binomials will yield an outside/inside term equal to the middle term.

Since signs are the same, take the sum: 3 4 7

( )( )x x

( 3)( 4)x x

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2 15 36x x

Example 5

( )( )x x

( )( )x x

1. Write the binomials with first terms

2. Determine the signs.

Third term positive – signs are the sameMiddle term positive – both are positive

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1 36

2 18

3 12

4 9

6 6

3. Determine the possible factors of the third term.

4. Find which pair of factors as last terms in the binomials will yield an outside/inside term equal to the middle term.

Since signs are the same, take the sum: 3 12 15

( )( )x x

( 3)( 12)x x

2 15 36x x

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