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19
THE DIFFERENCE
OF TWO SQUARES
WHEN THE SUM of two numbers multiplies their difference --
(a + b)(a − b)
-- then the product is the difference of their squares:
(a + b)(a − b) = a² − b²
For, the like terms will cancel. (Lesson 16.)
Symmetrically, the difference of two squares can be factored:
x² − 25 = (x + 5)(x − 5)
x² is the square of x. 25 is the square of 5.
Example 1. Multiply (x3 + 2)(x3 − 2).
Solution. Recognize the form: (a + b)(a − b). The product will be thedifference of two squares:
(x3 + 2)(x3 − 2) = x6 − 4.
When confronted with the form (a + b)(a − b), the student should notdo the FOIL method. The student should recognize immediately that the product will be a² − b².
Also, the order of factors never matters:
(a + b)(a − b) = (a − b)(a + b) = a² − b².
Problem 1. Write only final product..
| a) | (x + 9)(x − 9) = x² − 81 | b) | (y + z)(y − z) = y² − z² | |
| c) | (6x − 1)(6x + 1) = 36x² − 1 | d) | (3y + 7)(3y − 7) = 9y² − 49 | |
| e) | (x3 − 8)(x3 + 8) = x6 − 64 | f) | (xy + 10)(xy − 10) = x²y² − 100 | |
| g) | (xy² − z3)(xy² + z3) = x²y4 − z6 | h) | (xn + ym)(xn − ym) = x2n − y2m | |
Problem 2. Factor.
| a) | x² − 100 = (x + 10)(x − 10) | b) | y² − 1 = (y + 1)(y − 1) | |
| c) | 1 − 4z² = (1 + 2z)(1 − 2z) | d) | 25m² − 9n² = (5m + 3n)(5m − 3n) | |
| e) | x6 − 36 = (x3 + 6)(x3 − 6) | f) | y4 − 144 = (y² + 12)(y² − 12) | |
| g) | x8 − y10 = (x4 + y5) (x4 − y5) | h) | x2n − 1 = (xn + 1)(xn − 1) | |
Problem 3. Factor completely.
| a) x4 − y4 | = | (x² + y²)(x² − y²) |
| = | (x² + y²)(x + y)(x − y) | |
| b) 1 − z8 | = | (1 + z4)(1 − z4) |
| = | (1 + z4)(1 + z²)(1 − z²) | |
| = | (1 + z4)(1 + z²)(1 + z)(1 − z) | |
Problem 4. Completely factor each of the following. First remove acommon factor. Then factor the difference of two squares.
a) xy² − xz² = x(y² − z²) = x(y + z)(y − z)
b) 8x² − 72 = 8(x² − 9) = 8(x + 3)(x − 3)
c) 64z − z3 = z(64 − z²) = z(8 + z)(8 − z)
d) rs3 − r3s = rs(s² − r²) = rs(s + r)(s − r)
e) 32m²n − 50n3 = 2n(16m² − 25n²) = 2n(4m + 5n)(4m − 5n)
f) 5x4y5 − 5y5 = 5y5(x4 − 1) = 5y5(x² + 1)(x + 1)(x − 1)
Geometrical algebra
On the left, the entire figure is a square on side a. The square b² has been inserted in the lower right corner, so that the shaded area is the differenceof the two squares, a² − b².
Now, in the figure on the right, we have moved the rectangle b(a − b) to the top. The shaded area is now equal to the rectangle
(a + b)(a − b).
That is,
a² − b² = (a + b)(a − b).
*
The Difference of Two Squares completes our study of products of binomials. Those products come up so often that the student should be able to recognize and apply each form.
Summary of Multiplying/Factoring
In summary, here are the four forms of Multiplying/Factoring that characterize algebra.
 | ||||
| 1. Common Factor | 2(a + b) | = | 2a + 2b | |
| 2. Quadratic Trinomial | (x + 2)(x + 3) | = | x² + 5x + 6 | |
| 3. Perfect Square Trinomial | (x − 5)² | = | x² − 10x + 25 | |
| 4. The Difference of Two Squares | (x + 5)(x − 5) | = | x² − 25 | |
 | ||||
Problem 5. Distinguish each form, and write only the final product.
a) (x − 3)² = x² − 6x + 9. Perfect square trinomial.
b) (x + 3)(x − 3) = x² − 9. The difference of two squares.
c) (x − 3)(x + 5) = x² + 2x − 15. Quadratic trinomial.
d) (2x − 5)(2x + 5) = 4x² − 25. The difference of two squares.
e) (2x − 5)² = 4x² − 20x + 25. Perfect square trinomial.
f) (2x − 5)(2x + 1) = 4x² − 8x − 5. Quadratic trinomial.
Problem 6. Factor. (What form is it? Is there a common factor? Is it the difference of two squares? . . . )
a) 6x − 18 = 6(x − 3). Common factor.
b) x6 + x5 + x4 + x3 = x3(x3 + x² + x + 1). Common factor.
c) x² − 36 = (x + 6)(x − 6). The difference of two squares.
d) x² − 12x + 36 = (x − 6)². Perfect square trinomial.
e) x² − 6x + 5 = (x − 5)(x − 1). Quadratic trinomial.
f) x² − x − 12 = (x − 4)(x + 3)
g) 64x² − 1 = (8x + 1)(8x − 1)
h) 5x² − 7x − 6 = (5x + 3)(x − 2)
i) 4x5 + 20x4 + 24x3 = 4x3(x² + 5x + 6) = 4x3(x + 3)(x + 2)
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