Question

-2. The sum of two cubes can be factored by using the formula o’ + b3 (a + b)(c? ab + b?).(a) Verify the formula by multiplying the right side of the equation.(b) Factor the expression 8x2 + 27.(C) One of the factors of q? - bºis a - b. Find a quadratic factor of q? - bº. Show your work.(d) Factor the expression x - 1.

Answer 1

Given that the sum of two cubes can be factored by using the formula

`a^3+b^3=(a+b)(a^2-ab+b^2)`

a) To verify the formula by multiplying the right side equation

`\begin{gathered} (a+b)(a^2-ab+b^2) \\ =a(a^2-ab+b^2)+b(a^2-ab+b^2) \\ =a^3-a^2b+ab^2+a^2b-ab^2+b^3 \\ \text{Collect like terms} \\ =a^3-a^2b+a^2b+ab^2-ab^2+b^3 \\ \text{Simplify} \\ =a^3+b^3 \end{gathered}`

Hence,

`(a+b)(a^2-ab+b^2)=a^3+b^3`

b) To factor

`8x^3+27`

Using the sum of two cubes formula, i.e

`a^3+b^3=(a+b)(a^2-ab+b^2)`

Factorizing the expression gives

`\begin{gathered} (2x)^3+(3)^3=(2x+3)((2x)^2-(2x)(3)+(3)^2)_{} \\ (2x)^3+(3)^3=(2x+3)(4x^2-6x+9) \end{gathered}`

Hence, the answer is

`(2x+3)(4x^2-6x+9)`

c) Given that one of the factors of a³ - b³ is a- b, the quadratic factor of a³ - b³ can be deduced by applying the differences of cubes formula below

`a^3-b^3=(a-b)(a^2+ab+b^2)^{}_{}`

Expanding the right side equations

`\begin{gathered} (a-b)(a^2+ab+b^2)^{}_{}=a(a^2+ab+b^2)-b(a^2+ab+b^2) \\ =a^3+a^2b+ab^2-a^2b-ab^2-b^3 \\ \text{Collect like terms} \\ =a^3+a^2b-a^2b+ab^2-ab^2-b^3 \\ \text{Simplify} \\ =a^3-b^3 \end{gathered}`

Hence, the quadratic factor is

`a^3-b^3=(a-b)(a^2+ab+b^2)`

d) To factor the expression

`x^3-1`

By applying the differences of cubes formula

`a^3-b^3=(a-b)(a^2+ab+b^2)`

Factorizing the expression gives

`\begin{gathered} (x)^3-(1)^3=(x-1)(x^2+(x)(1)+1^2)^{}_{} \\ x^3-1^3=(x-1)(x^2+x+1) \end{gathered}`

Hence, the answer is

`(x-1)(x^2+x+1)`