1. Factoring Sums or Differences of Cubes
5.6
Factoring Sums or Differences of Cubes
Formulas for Factoring Sums or Differences of Cubes
Using the Formulas
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2. Formulas for Factoring Sums or Differences of Cubes
We have seen that a difference of two squares can always be factored, but a sum of two squares is usually prime. The difference or sum of two cubes can always be factored
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3. Factoring a Sum or Difference of Two Cubes
A3 + B3 = (A + B)(A2 – AB + B2);
A3 – B3 = (A – B)(A2 + AB + B2)
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4. Using the Formulas
When factoring a sum or difference of cubes, it can be helpful to remember that = 8, 33 = 27, 43 = 64, 53 = 125, 63 = 216, and so on. We say that 2 is the cube root of 8, 3 is the cube root of 27, and so on.
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5. Example Write an equivalent expression by factoring: x3 – 8. Solution
First observe that x3 – 8 = x3 – 23
From the formula:
A3 – B3 = (A – B)(A2 + AB + B2)
We have
x3 – 23 = (x – 2)(x2 + 2x + 22)
= (x – 2)(x2 + 2x + 4)
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6. Example Write an equivalent expression by factoring: b6 + 125y 3.
Solution
First observe that b y 3 = (b2)3 + (5y)3
From the formula:
A3 + B3 = (A + B)(A2 – AB + B2)
We have
(b2)3 + (5y)3 = (b2 + 5y)((b2)2 – 5yb2 + (5y)2)
= (b2 + 5y)(b4 – 5yb2 + 25y2)
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7. Example Write an equivalent expression by factoring: 2m5 + 2m2.
Solution
First factor out the largest common factor:
2m5 + 2m2 = 2m2(m3 + 1).
Now factor the sum of two cubes:
= 2m2(m + 1)(m2 – m + 1)
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8. Using Factoring Facts Sum of cubes: A3 + B3 = (A + B)(A2 – AB + B2)
Difference of cubes:
A3 – B3 = (A – B)(A2 + AB + B2)
Difference of squares:
A2 – B2 = (A + B)(A – B)
There is no formula for factoring a sum of two squares.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley