1. Factoring Perfect-Square Trinomials and Differences of Squares
5.5
Factoring Perfect-Square Trinomials and Differences of Squares
Perfect-Square Trinomials
Differences of Squares
More Factoring by Grouping
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2. Perfect-Square Trinomials
Consider the trinomial x2 +8x + 16
To factor it, we can look for factors of 16 that add to 8. These factors are 4 and 4 and the factorization is
x2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)2
Note that the result is the square of a binomial. x2 + 8x + 16 is called a perfect-square trinomial. A trinomial, recognized as a perfect-square trinomial can be quickly factored.
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3. To Recognize a Perfect-Square Trinomial
Two terms must be squares, such as A2 and B2.
There must be no minus sign before A2 or B2.
The remaining term must be 2AB or its
opposite, –2AB.
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4. Factoring a Perfect-Square Trinomial
A2 + 2AB + B2 = (A + B)2;
A2 – 2AB + B2 = (A – B)2
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5. Example Factor m2 + 10m + 25. Solution Using the formula:
A2 + 2AB + B2 = (A + B)2
we have:
m2+ 10m + 25 = m2 + 2(m)(5) + 52
= (m + 5) 2
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6. Example Factor 5x2 – 10xy + 5y2. Solution
5x2 – 10xy + 5y2 = 5(x2 – 2xy + y2)
= 5(x2 – 2(x)(y) + y2)
= 5(x – y) 2
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7. Differences of Squares
When an expression like x2 – 49 is recognized as a difference of two squares, we can reverse another pattern first seen in Section 5.2.
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8. Factoring a Difference of Two Squares
A2 – B2 = (A + B)(A – B)
To factor a difference of two squares, write the product of the sum and difference of the quantities being squared.
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9. Example Factor x2 – 49. Solution A2 – B2 = (A + B)(A – B)
x 2 – 7 2 = (x + 7)(x – 7)
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10. Example Factor 3m2n4 – 27n2. Solution 3m2n4 – 27n2 = 3n2(m2n2 – 9)
Factor out 3n2
= 3n2(mn + 3)(mn – 3)
Factor the difference of two squares
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11. More Factoring by Grouping
Sometimes, when factoring a polynomial with four terms, we may be able to factor further.
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12. Example Factor: y6 – 9y4 + 2y2 – 18. Solution y6 – 9y4 + 2y2 – 18
Factor out y2 – 9
y2 – 9 can be factored further
= (y4 + 2)(y – 3)(y + 3)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley