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# Factoring Trinomials

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

The product of two binomials can be a trinomial.

For example, letâ€™s multiply (x + 7) by (x + 3).

 Use FOIL. (x + 7)(x + 3) = x Â· x + x Â· 3 + 7 Â· x + 7 Â· 3 Simplify each term. Combine like terms. == x2x2 ++ 3x +10x 7x ++ 2121

Notice the relationship between the trinomial x2 + 10x + 21 and the binomials (x + 7) and (x + 3):

 â€¢ The first term of the trinomial, x2, is the product of x and x, the first term of each binomial. (x + 7)(x + 3) â€¢ The coefficient of the middle term of the trinomial is 10, the sum of 3 and 7, the constants in the binomials. = x Â· x + (7 + 3)x + 7 Â· 3 = x2 + 10x + 21 â€¢ The last term of the trinomial is 21, is the product of 3 and 7, the constants in the binomials.

This relationship holds in general. That is, if the product of two binomials (x + r) and (x + s) is a trinomial of the form x2 + bx + c:

â€¢ c is the product of r and s.

â€¢ b is the sum of r and s.

We use this to factor trinomials of the form x2 + bx + c.

This procedure is called the product-sum method of factoring because we seek two integers whose product is c and whose sum is b.

Procedure â€” To Factor x2 + bx + c (Product-Sum Method)

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

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

To check the factorization, multiply the binomial factors.

Example

Factor: x2 + 3x + 2

Solution

This trinomial has the form x2 + bx + c where b = 3 and c = 2.

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

Since c is 2, list pairs of integers whose product is 2.

Then, find the sum of each pair of integers.

 Product 1 Â· 2 (-1) Â· (-2) Sum3 -3

The first product, 1 Â· 2, gives the required sum, 3.

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 + 3x + 2 = (x + 1)(x + 2).

We multiply to check the factorization.

Is (x + 1)(x + 2) = x2 + 3x + 2 ?

Is x2 + 2x + 1x + 2 = x2 + 3x + 2 ?

Is x2 + 3x + 2 = x2 3x + 2 ? Yes

Note:

Multiplication is commutative, so the factorization may also be written: (x + 2)(x + 1).