Page 613 - Algebra 1

INVE9STIGATION
Choosing a Factoring Method
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Whole numbers can be factored into prime numbers. For example, 30 can be written as a product of its factors: 30 = 2 · 3 · 5. A factorization can be verified as correct by multiplying the factors and verifying that the product is the same as the original whole number. Similarly, many polynomials, as shown below, can also be factored.
x3y+2xy2 x3 +7x2 +2x+14 x2 -4 x2 +16x+64 x2 +8x+15 2x2 +7x+3
The difference between factoring whole numbers and polynomials is that often the factors of polynomials are other polynomials and not whole numbers. Factorization can be checked through multiplication and simplification. If the original polynomial results after correctly multiplying and simplifying, the polynomial has been factored correctly. If not, try again, or the polynomial may in fact be prime.
Many possibilities and methods exist for factoring. To begin the factoring process, it is helpful to have a checklist.
Checklist Item 1: Look for the greatest common factor. Does each term have a common factor?
For example, factor x3y + 2xy2.
1. Write each term of the polynomial as a product of its factors. 2. What does each term of the polynomial have in common?
3. Factor out the monomial from each term.
4. Now factor out the monomial from the polynomial.
Checklist Item 2: Look for a difference of two squares. Are there only two terms of the polynomial, and are they being subtracted? Are those two terms perfect squares?
For example, factor x2 - 4.
5. To begin the process of factoring, start with the first item on the checklist and proceed. Is there a common factor for all terms in the binomial?
6. Are there only two terms being subtracted?
7. The polynomial is a binomial of the form a2 - b2. What is the value of
a? b?
8. Use the factorization of the difference of two squares, a2 - b2 =
(a + b)(a - b), to factor the binomial.
9. If the two terms of the binomial were perfect squares being added, (x2 + 4), could that binomial be factored similarly?
598 Saxon Algebra 1