Complete 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:
[tex]A = B[/tex]
Step-by-step explanation:
See comment for complete question
Given
[tex]A: (4x^2 - 4x)(x^2 - 4)[/tex]
[tex]B: (x^2 + x - 2)(4x^2 - 8x)[/tex]
Required
Determine how they can show if the products are the same or not
To do this, we simply factorize each polynomial
For, Polynomial A: We have:
[tex]A: (4x^2 - 4x)(x^2 - 4)[/tex]
Factor out 4x
[tex]A: 4x(x - 1)(x^2 - 4)[/tex]
Apply difference of two squares on x^2 - 4
[tex]A: 4x(x - 1)(x - 2)(x+2)[/tex]
For, Polynomial B: We have:
[tex]B: (x^2 + x - 2)(4x^2 - 8x)[/tex]
Expand x^2 + x - 2
[tex]B:(x^2 + 2x - x - 2)(4x^2- 8x)[/tex]
Factorize:
[tex]B:(x(x + 2) -1(x + 2))(4x^2- 8x)[/tex]
Factor out x + 2
[tex]B:(x -1) (x + 2)(4x^2- 8x)[/tex]
Factor out 4x
[tex]B:(x -1) (x + 2)4x(x- 2)[/tex]
Rearrange
[tex]B: 4x(x - 1)(x - 2)(x+2)[/tex]
The simplified expressions are:
[tex]A: 4x(x - 1)(x - 2)(x+2)[/tex] and
[tex]B: 4x(x - 1)(x - 2)(x+2)[/tex]
Hence, both polynomials are equal
[tex]A = B[/tex]
