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 the difference of two squares:
(x3 + 2)(x3 − 2) = x6 − 4
x6 is the square of x3. 4 is the square of 2.
Problem 1. Write only final product..
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)
2nd Level
Next Lesson: Algebraic fractions
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