A. Factoring a Difference of Squares
Explore the Lesson
A. Factoring a Difference of Squares
In the
Warm Up, you saw that when the areas of two squares were subtracted, you were always left with the area of a shape that could be rearranged into a rectangle. In previous lessons, you looked at the relationship between the length, width, and
area of a rectangle and how they relate to multiplying and factoring polynomials. A very special combination of these ideas gives a method of factoring polynomials that are written as a
difference of
squares.
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Difference of Squares
a perfect square subtracted from another perfect square |
From the sequence of diagrams above, you can see that the original red area,
, minus the black area,
, is equal to the area in the final rectangle, whose length and width are
and
, respectively. From this result, it can be concluded that
. These findings give a method for factoring binomials whose terms are a difference of perfect squares. Before continuing with the lesson, try to factor
.
Key Lesson Marker
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Example 1 |
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Factor
Both
The factors can be written as the sum and the difference of these two square roots.
These factors can be verified through multiplication.
|
.
and 25 are perfect squares, so
is a difference of squares. Begin by determining the positive square root of each term.
