Question

Expand or factor each of the following expressions to determine which expressions are equivalent.​

Answer 1

(3x-4y)^2=9x^2-24x+16y^2
(3x-2)(9x^2+6x+4)=27x^3-8

Answer 2

The pair of equivalent expressions are 2 and 8, 3 and 7, 4 and 5.

Explanation:

Expand or factor each of the following expressions

1.

`(4x-3y)^2=(4x)^2-(2(4x)(3y)+(3y)^2`
`[\because (a-b)^2=a^2-2ab+b^2]`

`(4x-3y)^2=16x^2-24xy+9y^2`

2.

`9x^2+3x-20=9x^2+15x-12x-20`

`9x^2+3x-20=3x(3x+5)-4(3x+5)`

`9x^2+3x-20=(3x+5)(3x-4)`

It is same as expression 8. Therefore expression 2 and 8 are equivalent.

3.

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

`(3x-2)(9x^2+6x+4)=(3x)^3-(2)^3`
`[\because a^3-b^3=(a-b)(a^2+ab+b^2)]`

`(3x-2)(9x^2+6x+4)=27x^3-8`

It is same as expression 7. Therefore expression 3 and 7 are equivalent.

4.

`9x^2-24xy+16y^2=(3x)^2-2(3x)(4y)+(4y)^2`

`9x^2-24xy+16y^2=(3x-4y)^2`
`[\because (a-b)^2=a^2-2ab+b^2]`

It is same as expression 5. Therefore expression 4 and 5 are equivalent.

5.

`(3x-4y)^2=9x^2-24xy+16y^2`

6.

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

`(3x+2)(9x^2-6x+4)=(3x)^3+(2)^3`
`[\because a^3+b^3=(a+b)(a^2-ab+b^2)]`

`(3x+2)(9x^2-6x+4)=27x^3+8`

7.

`27x^3-8=(3x-2)(9x^2+6x+4)`

8.

`(3x+5)(3x-4)=9x^2+3x-20`

Therefore the pair of equivalent expressions are 2 and 8, 3 and 7, 4 and 5.