Question

day 3 – sum and difference of cubes
factor please.

1. $x^3 + 125$
2. $a^3 - 64$
3. $x^3 - 216$
4. $y^3 + 27$
5. $8 - 27m^3$
6. $125x^3 + 8$
7. $8x^3 - y^3$
8. $27y^3 + 64x^3$
9. $3 + 3a^3$
10. $4 - 4u^3$
11. $108 - 4x^3$
12. $216a^3 + 125b^3$

Explanation:

Key Formulas:

Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

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1. Factor $x^3 + 125$

Step1: Identify cube terms

$x^3 = x^3$, $125 = 5^3$

Step2: Apply sum of cubes

$(x + 5)(x^2 - 5x + 5^2)$

Step3: Simplify constant term

$(x + 5)(x^2 - 5x + 25)$

2. Factor $a^3 - 64$

Step1: Identify cube terms

$a^3 = a^3$, $64 = 4^3$

Step2: Apply difference of cubes

$(a - 4)(a^2 + 4a + 4^2)$

Step3: Simplify constant term

$(a - 4)(a^2 + 4a + 16)$

3. Factor $x^3 - 216$

Step1: Identify cube terms

$x^3 = x^3$, $216 = 6^3$

Step2: Apply difference of cubes

$(x - 6)(x^2 + 6x + 6^2)$

Step3: Simplify constant term

$(x - 6)(x^2 + 6x + 36)$

4. Factor $y^3 + 27$

Step1: Identify cube terms

$y^3 = y^3$, $27 = 3^3$

Step2: Apply sum of cubes

$(y + 3)(y^2 - 3y + 3^2)$

Step3: Simplify constant term

$(y + 3)(y^2 - 3y + 9)$

5. Factor $8 - 27m^3$

Step1: Identify cube terms

$8 = 2^3$, $27m^3 = (3m)^3$

Step2: Apply difference of cubes

$(2 - 3m)(2^2 + 2(3m) + (3m)^2)$

Step3: Simplify all terms

$(2 - 3m)(4 + 6m + 9m^2)$

6. Factor $125x^3 + 8$

Step1: Identify cube terms

$125x^3 = (5x)^3$, $8 = 2^3$

Step2: Apply sum of cubes

$(5x + 2)((5x)^2 - (5x)(2) + 2^2)$

Step3: Simplify all terms

$(5x + 2)(25x^2 - 10x + 4)$

7. Factor $8x^3 - y^3$

Step1: Identify cube terms

$8x^3 = (2x)^3$, $y^3 = y^3$

Step2: Apply difference of cubes

$(2x - y)((2x)^2 + (2x)(y) + y^2)$

Step3: Simplify all terms

$(2x - y)(4x^2 + 2xy + y^2)$

8. Factor $27y^3 + 64x^3$

Step1: Identify cube terms

$27y^3 = (3y)^3$, $64x^3 = (4x)^3$

Step2: Apply sum of cubes

$(3y + 4x)((3y)^2 - (3y)(4x) + (4x)^2)$

Step3: Simplify all terms

$(3y + 4x)(9y^2 - 12xy + 16x^2)$

9. Factor $3 + 3a^3$

Step1: Factor out GCF

$3(1 + a^3)$

Step2: Identify cube terms

$1 = 1^3$, $a^3 = a^3$

Step3: Apply sum of cubes

$3(1 + a)(1^2 - 1(a) + a^2)$

Step4: Simplify inner terms

$3(1 + a)(1 - a + a^2)$

10. Factor $4 - 4u^3$

Step1: Factor out GCF

$4(1 - u^3)$

Step2: Identify cube terms

$1 = 1^3$, $u^3 = u^3$

Step3: Apply difference of cubes

$4(1 - u)(1^2 + 1(u) + u^2)$

Step4: Simplify inner terms

$4(1 - u)(1 + u + u^2)$

11. Factor $108 - 4x^3$

Step1: Factor out GCF

$4(27 - x^3)$

Step2: Identify cube terms

$27 = 3^3$, $x^3 = x^3$

Step3: Apply difference of cubes

$4(3 - x)(3^2 + 3(x) + x^2)$

Step4: Simplify inner terms

$4(3 - x)(9 + 3x + x^2)$

12. Factor $216a^3 + 125b^3$

Step1: Identify cube terms

$216a^3 = (6a)^3$, $125b^3 = (5b)^3$

Step2: Apply sum of cubes

$(6a + 5b)((6a)^2 - (6a)(5b) + (5b)^2)$

Step3: Simplify all terms

$(6a + 5b)(36a^2 - 30ab + 25b^2)$

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Answer:

1. $(x + 5)(x^2 - 5x + 25)$
2. $(a - 4)(a^2 + 4a + 16)$
3. $(x - 6)(x^2 + 6x + 36)$
4. $(y + 3)(y^2 - 3y + 9)$
5. $(2 - 3m)(4 + 6m + 9m^2)$
6. $(5x + 2)(25x^2 - 10x + 4)$
7. $(2x - y)(4x^2 + 2xy + y^2)$
8. $(3y + 4x)(9y^2 - 12xy + 16x^2)$
9. $3(1 + a)(1 - a + a^2)$
10. $4(1 - u)(1 + u + u^2)$
11. $4(3 - x)(9 + 3x + x^2)$
12. $(6a + 5b)(36a^2 - 30ab + 25b^2)$