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.
• Since the product, c = 6, is positive, the integers must have the same
sign.
• Also, the sum, b = -5, is negative. So the integers must both be
negative.
| Product
-1 · -6
-2 · -3 |
Sum
-7
-5 |
The last possibility, -2 · (-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 ? |
| Is Is
Is |
x · x
x2
x2 |
+ +
- |
x · (-3y)
(-3xy)
|
+ +
5xy |
(-2y) · x
(-2xy)
|
+ +
+ |
(-2y) · (-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.
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