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# Factoring Trinomials

## Factoring a Trinomial of the Form x2 + bxy + cy2

A trinomial of the form x2 + bxy + cy2 contains two variables, x and y.

Notice that the middle term has factor y and the last term has factor of y2.

To factor x2 + bxy + cy2, we may follow the same procedure we used for x2 + bx + c. However, we include the variable y in the second term of each binomial factor.

Example 1

Factor: x2 - 5xy + 6y2

Solution

This binomial has the form x2 + bxy + cy2 where b = -5 and c = 6.

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

â€¢ Since the product, c = 6, is positive, the integers must have the same sign.

â€¢ Also, the sum, b = -5, is negative. So the integers must both be negative.

 Product -1 Â· -6 -2 Â· -3 Sum -7 -5
The last possibility, -2 Â· (-3), gives the required sum, -5.

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

The result is: x2 - 5xy + 6y2 = (x - 2y)(x - 3y).

Note that each binomial has y in its second term.

We can multiply to check the factorization.

 Is (x - 2y)(x - 3y) = x2 - 5xy + 6y2 ? IsIs Is x Â· x x2 x2 ++ - x Â· (-3y) (-3xy) ++ 5xy (-2y) Â· x (-2xy) ++ + (-2y) Â· (-3y) 6y2 6y2 = x2 - 5xy + 6y2 ?= x2 - 5xy + 6y2 ? = x2 - 5xy + 6y2 ? Yes

Example 2

Factor: x2 - 5x + 6y2

Solution

This trinomial does not have the form x2 + bxy + cy2 because there is no factor y in the middle term. This trinomial cannot be factored using integers.