Question

Find the following quotients, and write the quotient in standard form. (a)- x^2-9 ÷ x-3 (b)- x^3-27 ÷ x-3 (c) x^4- 81 ÷ x-3

Answer 1

(a) x+3

(b)
`x^2+3x+9`

(c)
`(x+3)(x^2+9)`

Explanation:

We have given that

(a)
`(x^2-9)/(x-3)`

From the algebraic identity we know that

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

So
`(x^2-9)/(x-3)=((x+3)(x-3))/(x-3)=x+3`

(b)
`(x^3-27)/(x-3)`

We know the algebraic identity

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

So
`(x^3-27)/(x-3)=(x^3-3^3)/(x-3)=((x-3)(x^2+3x+9))/(x-3)=x^2+3x+9`

(c) We have given
`(x^4-81)/(x-3)`

We know the algebraic identity

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

`(x^4-81)/(x-3)=((x^2)^2-(3^2)^2)/(x-3)=((x^2-9)(x^2+9))/(x-3)=((x+3)(x-3)(x^2+9))/(x-3)=(x+3)(x^2+9)`