Factoring

Difference of Squares

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A binomial is factorable only if it is one of three things a Difference of Squares, a Difference of Cubes, or a Sum of Cubes. A binomial is a Difference of Squares if both terms are perfect squares. Recall we may have to factor out a common factor first.

If we determine that a binomial is a difference of squares, we factor it into two binomials. The first being the square root of the first term minus the square root of the second term. The second being the square root of the first term plus the square root of the second term, as in the following formula:

 A2 - B2 = (A - B)(A + B)

Factor each of the following.

1. x2 - 25: solution

2. 16x4 - 100 : solution

3. 50x2 - 72: solution

#1 solution

1. x2 - 25

first we check that the binomial is a difference of squares.
x2 is a perfect square, and so is 25, so yes we have a difference of squares.
the square root of x2 is x, and the square root of 25 is 5
so our answer is (x - 5)(x + 5)

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#2 solution

2. 16x4 - 100

16x4 is a perfect square, as is 100, so we do have a difference of squares
Here we must first factor out the common factor, if we do not our answer will not be completely factored.
Our common factor is 4, giving us 4(4x4 - 25)
the square root of 4x4 is 2x2, the square root of 25 is 5.
so our answer is 4(2x2 - 5)(2x2 + 5)

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#3 solution

3. 50x2 - 72

Here neither 50x2 nor 72 are perfect squares, but we must first take out the common factor
the common factor is 2, giving us 2(25x2 - 36)
Now both 25x2 and 36 are perfect squares so we have a difference of squares.
The square root of 25x2 is 5x and the square root of 36 is 6.
so our answer is 2(5x - 6)(5x + 6)

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