Factoring Trinomials
Factoring a Trinomial of the Form x^{2} + bxy + cy^{2}
A trinomial of the form x^{2} + bxy + cy^{2} contains two variables, x and y.
Notice that the middle term has factor y and the last term has factor of y^{2}.
To factor x^{2} + bxy + cy^{2}, we may follow the same procedure we used for
x^{2} + bx + c. However, we include the variable y in the second term of
each binomial factor.
Example 1
Factor: x^{2}  5xy + 6y^{2}
Solution
This binomial has the form x^{2} + bxy + cy^{2} 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:
x^{2}  5xy + 6y^{2} = (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) 
= x^{2}  5xy + 6y^{2} ? 
Is Is
Is 
x Â· x
x^{2}
x^{2} 
+ +
 
x Â· (3y)
(3xy)

+ +
5xy 
(2y) Â· x
(2xy)

+ +
+ 
(2y) Â· (3y)
6y^{2}
6y^{2} 
= x^{2}  5xy + 6y^{2} ? = x^{2}
 5xy + 6y^{2} ?
= x^{2}  5xy + 6y^{2} ? Yes 
Example 2
Factor: x^{2}  5x + 6y^{2}
Solution
This trinomial does not have the form x^{2} + bxy + cy^{2} because there is no
factor y in the middle term. This trinomial cannot be factored using
integers.
