Factoring Polynomials 137
Guidelines for Factoring
Finally, here are some general guidelines for factoring of polynomials.
- Count the number of terms.
- If the expression has a GCF, factor out the GCF.
- If there are two terms, check for a special binomial product.
- If there are three terms, check for a quadratic trinomial.
- If there are four terms, try grouping in pairs.
- Check whether any previously obtained factor can be factored further.
Problem Factor completely.
a. 100 x^4 y^2 z − 25 x^2 y^2 z
b. x^2 (x + y) + 2 xy(x + y) + y^2 (x + y)
Solution
a. 100 x^4 y^2 z − 25 x^2 y^2 z
Step 1. Factor out the GCF, 25x^2 y^2 z.
100 x^4 y^2 z − 25 x^2 y^2 z
= 25 x^2 y^2 z ⋅ 4 x^2 − 25 x^2 y^2 z ⋅ 1
= 25 x^2 y^2 z(4x^2 − 1)
Step 2. Factor the difference of two squares, 4x^2 − 1.
25 x^2 y^2 z(2x− 1)(2x+ 1)is the completely factored form.
b. x^2 (x + y) + 2 xy (x + y) + y^2 (x + y)
Step 1. Factor out the GCF, (x + y).
x^2 (x + y) + 2 xy (x + y) + y^2 (x + y)
()++ ()++++
Step 2. Factor the perfect trinomial square, x^2 + 2 xy+y^2 , and simplify.
()xy+ ()xy+ =()
23
() is the completely factored form.
