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## Factoring a Trinomial of the Form x2 + bx + c

Example 1

Factor: x2 - 7x + 12

Solution

This trinomial has the form x2 + bx + c where b = -7 and c = 12.

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

Since c is 12, list pairs of integers whose product is 12. Then, find the sum of each pair of integers.

 Product 1 · 12 2 · 6 3 · 4 -1 · (-12) -2 · (-6) -3 · (-4) Sum13 8 7 -13 -8 -7

The last possibility, -3 · (-4), gives the required sum, -7.

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 - 7x + 12 = (x - 3)(x - 4).

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

Note:

The product, c = 12, is positive, so both integers are positive or both are negative.

Since we also know the sum, b = -7, is negative, we can conclude that both integers are negative.

So we did not have to try the positive integers.

Example 2

Factor: x2 + x - 30

Solution

This trinomial has the form x2 + bx + c where b = 1 and c = -30.

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

There are eight possible integer pairs whose product is -30.

To reduce the list, think about the signs of 1 and -30.

 Since the product, c = -30, is negative, one factor must be positive and the other negative.

 Also, the sum, b = 1, is positive. So the integer with the greater absolute value must be positive. We need only list pairs of integers whose sum is positive.

 Product -1 · 30 -2 · 15 -3 · 10 -5 · 6 Sum 29 13 7 1
The last possibility, -5 · 6, gives the required sum, 1.

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 + x - 30 = (x - 5)(x + 6).

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

Note:

These are the eight integer pairs with product -30:

-1, 30

-2, 15

-3, 10

-5, 6

1, -30

2, -15

3, -10

5, -6

Only one pair, -5 and 6, gives the required sum, 1.