The standard form of any quadratic trinomial is

Standard Form

cbxax ++2

€

So, in 3x 2 − 4x + 1... a=3

b=-4

c=1

Now you try.

€

−x 2 + 7x − 2a = b = c =

€

2x 2 − x + 5

a = b = c =

€

4x 2 + x − 2

a = b = c =

Factoring when a=1 and c > 0.

First list all the factor pairs of c.

€

x 2 + 8x + 12

Then find the factors with a sum of b

These numbers are used to make the factored expression.

€

x + 2( ) x + 6( )

1 , 122 , 63 , 4

Now you try.

€

x 2 + 8x + 15

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x 2 + 10x + 21Factors of c: Factors of c:

Binomial Factors

( ) ( )

Circle the factors of c with the sum of b

Circle the factors of c with the sum of b

Binomial Factors

( ) ( )

Factoring when c >0 and b < 0.

c is positive and b is negative.Since a negative number times a negative number

produces a positive answer, we can use the same method as before but…

The binomial factors will have subtraction instead of addition.

Let’s look at

€

x 2 − 13x + 12

1 12

2 6

3 4

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x − 12( ) x − 1( )

We need a sum of -13

Make sure both values are negative!

First list the factors of 12

Now you try.

€

1. x 2 − 7x +12

€

2. x 2 − 9x +14

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3. x 2 −13x + 42

Factoring when c < 0.

We still look for the factors of c. However, in this case, one factor should be

positive and the other negative in order to get a negative value for c

Remember that the only way we can multiply two numbers and

come up with a negative answer, is if one is number is positive and the other is negative!

Let’s look at

€

x 2 − x − 12

1 12

2 6

3 4

In this case, one factor should be positive and the other negative.

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x + 3( ) x − 4( )

We need a sum of -1 + -

Another Example

€

x 2 + 3x − 18

1 18

2 9

3 6

List the factors of 18.

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x − 3( ) x + 6( )

We need a sum of 3

What factors and signswill we use?

Now you try.

€

x 2 + 3x − 4

x 2 + x − 20

x 2 − 4x − 21

x 2 − 10x − 56

1.

2.

3.

4.

Prime Trinomials

Sometimes you will find a quadratic trinomial that is not

factorable. You will know this when

you cannot get b from the list of factors.

When you encounter this write not factorable or

prime.

Here is an example…

€

x 2 + 3x + 18 1 18

2 9

3 6

Since none of the pairs adds to 3, this trinomial is prime.

Now you try.

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x 2 − 6x + 4

€

x 2 − 10x − 39

€

x 2 + 5x − 7

factorable prime

factorable prime

factorable prime

When a ≠ 1.

Instead of finding the factors of c:Multiply a times c.Then find the factors of this product.

€

7x 2 − 19x + 10

a × c = 70

1 70

2 35

5 14

7 10

1 70

2 35

5 14

7 10

We still determine the factors that add to b.

So now we have

But we’re not finished yet….

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x − 5( ) x − 14( )

Since we multiplied in the beginning, we need to divide in the end.

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x −5

7

⎛

⎝ ⎜

⎞

⎠ ⎟ x −

14

7

⎛

⎝ ⎜

⎞

⎠ ⎟

€

x −5

7

⎛

⎝ ⎜

⎞

⎠ ⎟ x − 2( )

Divide each constant by a.Simplify, if possible.

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7x − 5( ) x − 2( )Clear the fraction in each binomial factor

Recall

• Divide each constant by a.

• Simplify, if possible.

• Clear the fractions.

• Multiply a times c.

• List factors. Look for sum of b• Write 2 binomials using the factors with sum of b

932 2 −− xx

€

2 × 9 =18

€

1 18

2 9

3 6

€

x − 6( ) x + 3( )

€

x −6

2

⎛

⎝ ⎜

⎞

⎠ ⎟ x +

3

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

x − 3( ) x +3

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

x − 3( ) 2x + 3( )

Now you try.

€

4x 2 + 4x − 3

€

6x 2 − 23x + 7€

3x 2 − 5x −12

Sometimes there is a GCF.

If so, factor it out first.

Then use the previous methods to factor the trinomial

€

4x2 − 2x − 30

€

2 2x 2 − x −15( )

€

2 x − 6( ) x + 5( )

€

2 x −6

2

⎛

⎝ ⎜

⎞

⎠ ⎟ x −

5

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

2 x − 3( ) x −5

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

2 x − 3( ) 5x + 2( )

Now you try.

€

4x 2 + 16x + 12

6x 2 + 10x + 6

1.

2.

Recall

First factor out the GCF.

€

45x2 − 35x −10

€

5 9x 2 − 7x − 2( )

€

5 x + 2( ) x − 9( )

€

5 x +2

9

⎛

⎝ ⎜

⎞

⎠ ⎟ x −

9

9

⎛

⎝ ⎜

⎞

⎠ ⎟

€

5 9x + 2( ) x −1( )

€

5 9x + 2( ) x −1( )

Then factor the remaining trinomial.

€

6x 2 + 30x − 36

€

4x 2 + 14x + 10

1.

2.