Question

Emily and Zach have two different polynomials to multiply:

Polynomial product A:
(4x2 – 4x)(x2 – 4)Polynomial product B:
(x2 + x – 2)(4x2 – 8x)

They are trying to determine if the products of the two polynomials are the same. But they disagree about how to solve this problem.

Answer 1

For multiplying two polynomials, we will multiply each term in the first parenthesis with the whole second parenthesis part and then use distributive property and simplify in the end.

Polynomial product A

`(4x^2 -4x)(x^2 -4)\\ \\ =4x^2(x^2-4)-4x(x^2-4)\\ \\ = 4x^4-16x^2-4x^3+16x\\ \\ =4x^4-4x^3-16x^2+16x`

Polynomial product B

`(x^2+x-2)(4x^2-8x)\\ \\ =x^2(4x^2-8x)+x(4x^2-8x)-2(4x^2-8x)\\ \\ =4x^4-8x^3+4x^3-8x^2-8x^2+16x\\ \\=4x^4-4x^3-16x^2+16x`

Thus, the products of the two polynomials are the same.

Answer 2

We need to multiply of given polynomials

Given:-

Polynomial product A :
`(4x^(2)-4x)(x^(2)-4)`

Polynomial product B :
`(x^(2)+x-2)(4x^(2)-8x)`

For product A :
`(4x^(2)-4x)(x^(2)-4)`

multiply term wise,

`4x^(2)(x^(2)-4)-4x(x^(2)-4)`

`4x^(2+2)-4* 4x^(2)-4x^(2+1)+4* 4x`

`4x^(4)-16x^(2)-4x^(3)+16x`

`4x^(4)-4x^(3)-16x^(2)+16x`

For product B :
`(x^(2)+x-2)(4x^(2)-8x)`

multiply term wise,

`x^(2)(4x^(2)-8x)+x(4x^(2)-8x)-2(4x^(2)-8x)`

`4x^(2+2)-8x^(2+1)+4x^(2+1)-8x^(1+1)-8x^(2)+16x`

`4x^(4)-4x^(3)-8x^(2)-16x^(2)+16x`

Therefore, the products of the two polynomials are the same