Factoring the Difference of Two Squares
A first-degree polynomial in one variable, such as 3x - 5, is called a linear polynomial.
(The equation 3x - 5 = 0 is a linear equation.)
Linear Polynomial
If a and b are real numbers with a ≠ 0, then ax
+ b is called a linear polynomial.
Asecond-degree polynomial such as x2 + 5x - 6 is called a quadratic polynomial.
Quadratic Polynomial
If a, b, and c are real numbers with a ≠ 0, then ax2
+ bx + c is called a
quadratic polynomial.
Helpful hint
The prefix “quad†means four.
So why is a polynomial of
three terms called quadratic?
Perhaps it is because a quadratic
polynomial can often
be factored into a product of
two binomials.
One of the main goals of this chapter is to write a quadratic polynomial (when
possible) as a product of linear factors.
Consider the quadratic polynomial x2 - 25. We recognize that x2
- 25 is a difference
of two squares, x2 - 52. We recall that the product of a sum and a difference
is a difference of two squares: (a + b)(a - b) = a2 - b2. If we reverse this special
product rule, we get a rule for factoring the difference of two squares.
Factoring the Difference of Two Squares
a2 - b2 = (a + b)(a - b)
The difference of two squares factors as the product of a sum and a difference. To
factor x2 - 25, we replace a by x and b by 5 to get
x2 - 25 = (x - 5)(x - 5).
This equation expresses a quadratic polynomial as a product of two linear factors.
Example 1
Factoring the difference of two squares
Factor each polynomial.
a) y2 - 36
b) 9x2 - 1
c) 4x2 - y2
Solution
Each of these binomials is a difference of two squares. Each binomial factors into a
product of a sum and a difference.
a) y2 - 36 = (y + 6)(y - 6) We could also write (y - 6)(y + 6) because
the factors can be written in any order.
b) 9x2 - 1 = (3x + 1)(3x - 1)
c) 4x2 - y2 = (2x + y)(2x - y)
Helpful hint
Using the power of a power
rule, we can see that any even
power is a perfect square:
x2n = (xn)2