factor
2x² + 9xy + 10y²
1. Multiply the 2 by the 10, get 20.
2. Notice the sign of the last term is +, so we think "SUM".
(If the sign of the last term had been -, we would think "DIFFERENCE")
3. Think of two positive integers which have product 20 and SUM 9, the
coefficient (in absolute value) of the middle term.
It doesn't take long to see that two such positive integers are 5 and 4.
4. Now use 5 and 4 to rewrite the 9 as (5 + 4)
2x² + (5 + 4)xy + 10y²
5. Remove the parentheses by distributing
2x² + 5xy + 4xy + 10y²
6. Factor by grouping. That is,
a. factor the first two terms by taking
out x.
b. factor the last two terms by taking out 2y
x(2x + 5y) + 2y(2x + 5y)
c. Now notice that (2x + 5y) is contained in both expressions:
x(2x + 5y) + 2y(2x + 5y)
d. So factor out the whole (2x + 5y) leaving x when factoring
it out of the left expression, and leaving 2y when factoring
(2x + 5y) out on the right expression.
(2x + 5y)(x + 2y)
That's it!
Edwin
