The difference of squares strategy can also be used with more complex expressions.

Example 2

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»36«/mn»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»100«/mn»«msup»«mi»e«/mi»«mn»2«/mn»«/msup»«/math».

Both «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»36«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»d«/mi»«mn»2«/mn»«/msup»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»100«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»e«/mi»«mn»2«/mn»«/msup»«/math» are perfect squares. Look at the coefficients and variables separately to determine the positive square roots.

The factors can be written as the sum and the difference of these two square roots.

These factors can be verified through multiplication.



Symbolically factoring a difference of squares using the relationship «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»)«/mo»«mo»(«/mo»«mi»x«/mi»«mo»-«/mo»«mi»y«/mi»«mo»)«/mo»«/math» is not the only possible method. A difference of squares can also be factored concretely and pictorially using algebra tiles and symbolically using the decomposition method learned in Lesson 5.3. For these methods, it is important to recognize that «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/math» can be thought of as a trinomial with a b-value of 0.

Example 3

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«/math» using algebra tiles.

A rectangle can be formed by introducing two pairs of x and − x tiles.


Example 4

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«msup»«mi»n«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»1«/mn»«/math» by decomposition.

ac = 4 and b = 0

The integers 2 and −2 have a product of −4 and a sum of 0.