Algebra Demystified 2nd Ed

Chapter 6 FaCtoring and the distributive ProPerty 159

The shortcut for factoring a quadratic polynomial where first term is x^2 can

help you identify quadratic polynomials that do not factor “nicely.” The next

three examples are quadratic polynomials that do not factor “nicely.”

x^2 + x + 1

x^2 + 14x + 19

x^2 – 5x + 10

Factoring the Difference of Two Squares

A quadratic polynomial of the form x^2 – c^2 is called the difference of two squares.

We can use the shortcut described above on x^2 – c^2 = x^2 + 0x – c^2. The factors

of c^2 must have a difference of 0. This can only happen if they are the same, so

the factors of c^2 we want are c and c, as shown in xc^22 −=−+()xc()xc.

EXAMPLES

Use the formula to factor the difference of two squares.

x^2 – 9 = (x – 3)(x + 3)

x^2 –100 = (x – 10)(x + 10)

x^2 – 49 = (x – 7)(x + 7)

16 – x^2 = (4 – x)(4 + x)

When the sign between x^2 and c^2 is plus, the quadratic cannot be factored

using real numbers. For example, the quadratic polynomial x^2 + 25 cannot

be factored using real numbers.

PRACTICE

Use the formula to factor the difference of two squares.

1. x^2 − 4 =


2. x^2 − 81 =


3. x^2 − 25 =


4. x^2 − 64 =


5. x^2 − 1 =


6. x^2 − 15 =


7. 25 − x^2 =

EXAMPLES

Use the formula to factor the difference of two squares.

PRACTICE

Use the formula to factor the difference of two squares.