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Factoring Trinomials

Factoring a Trinomial of the Form x2 + bx + c

Example 1

Factor: x2 + 2x + 4

Solution

This trinomial has the form x2 + bx + c where b = 2 and c = 4.

Step 1 Find two integers whose product is c and whose sum is b.

• Since the product, c = 4, is positive, both integers must have the same sign.

• Also the sum, b = 2, is positive. So both integers must be positive.

Product

1 · 4

2 · 2

Sum

5

4

These are the only possibilities, and neither gives the required sum, 2.

Since there are no two integers with product 4 and sum 2, the trinomial x2 + 2x + 4 cannot be factored as (x + r)(x + s) where r and s are integers.

Note:

It is possible to factor x2 + 2x + 4 using numbers other than integers, but that is beyond the scope of this lesson.

 

Example 2

Factor: x2 - 9

Solution

First we write the binomial x2 - 9 in the form x2 + bx + c.

To do this, we insert a middle term, 0x.

The equivalent trinomial is x2 + 0x - 9.

Now, we see that b = 0 and c = -9.

Step 1 Find two integers whose product is c and whose sum is b.

• Since the product, c = -9, is negative, one factor must be positive and the other negative.

• Since the sum is b = 0, the integers must be opposites.

There is only one possibility.

Product

-3 · 3

Sum

0

 

Step 2 Use the integers from Step 1 as the constants, r and s, in the binomial factors (x + r) and (x + s).

The result is: x2 - 9 = (x + 3)(x - 3).

You can multiply to check the factorization. We leave the check to you.

Note:

These are the three integer pairs with product -9:

1, -9

-1, 9

-3, 3

Only one pair, -3 and 3, gives the required sum, 0.

Recall that the product of (x - 3) and (x + 3) is the difference of two squares, this case x2 - 9.

 

Example 3

Factor: 3x2 - 6x - 9

Solution

This trinomial does not have the form x2 + bx + c because the coefficient of x2 is not 1. However, the terms of this trinomial have a GCF of 3.   3x2 - 6x - 9
Factor out the common factor, 3.

=

3(x2 - 2x - 3)
 

Now, the trinomial has the form x2 + bx + c where b = -2 and c = -3.

Step 1 Find two integers whose product is c and whose sum is b.

• Since the product, c = -3, is negative, one integer must be positive and the other negative.

• Also, the sum, b = -2, is negative. So the integer with the greater absolute value must be negative.

There is only one possibility.

Product

1 · -3

Sum

-2

 

Step 2 Use the integers from Step 1 as the constants, r and s, in the binomial factors (x + r) and (x + s).

The factorization of x2 - 2x - 3 is (x + 1)(x - 3).

But don’t forget the factor 3 that we factored out of the original trinomial.

The factorization of 3x2 - 6x - 9 is:

3x2 - 6x - 9 = 3(x + 1)(x - 3).

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