Page 155 - MarceAlgebra Demystified

142 CHAPTER 6 Factoring

This shortcut can help you identify quadratic polynomials that do not factor
‘‘nicely’’ without spending too much time on them. The next three examples
are quadratic polynomials that do not factor ‘‘nicely.’’
2
2
2
x þ x þ 1 x þ 14x þ 19 x 5x þ 10
2
2
Quadratic polynomials of the form x c are called the difference of two
2
2
2
2
squares. We can use the shortcut on x c ¼ x þ 0x c . The factors of
2
c must have a difference of 0. This can only happen if they are the same, so
2
the factors of c we want are c and c.
Examples
2 2
x 9 ¼ðx 3Þðx þ 3Þ x 100 ¼ðx 10Þðx þ 10Þ
2 2
x 49 ¼ðx 7Þðx þ 7Þ 16 x ¼ð4 xÞð4 þ xÞ
2
2
When the sign between x and c is plus, the quadratic cannot be factored
using real numbers.
Practice

2
1: x 4 ¼
2
2: x 81 ¼
2
3: x 25 ¼
2
4: x 64 ¼
2
5: x 1 ¼
2
6: x 15 ¼
2
7: 25 x ¼

Solutions

2
1: x 4 ¼ðx 2Þðx þ 2Þ
2
2: x 81 ¼ðx 9Þðx þ 9Þ