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Which expression is equivalent to the following polynomial expression?

[tex]\[ \left(2 x^5 + 3 y^4\right)\left(-4 x^2 + 9 y^4\right) \][/tex]

A. [tex]\(-2 x^{10} + 11 x^5 y^4 - x^2 y^4 + 12 y^{16}\)[/tex]

B. [tex]\(-8 x^7 + 27 y^8\)[/tex]

C. [tex]\(-8 x^7 + 18 x^5 y^4 - 12 x^2 y^4 + 27 y^8\)[/tex]

D. [tex]\(-2 x^7 + 11 x^5 y^4 - x^2 y^4 + 12 y^8\)[/tex]



Answer :

GinnyAnswer
Let's expand the given polynomial expression step-by-step:

We have:
[tex]\[ (2x^5 + 3y^4)(-4x^2 + 9y^4) \][/tex]

Using the distributive property (also known as the FOIL method for binomials), we must multiply each term in the first polynomial by each term in the second polynomial.

1. Multiply [tex]\(2x^5\)[/tex] by [tex]\(-4x^2\)[/tex]:
[tex]\[ 2x^5 \cdot (-4x^2) = -8x^{5+2} = -8x^7 \][/tex]

2. Multiply [tex]\(2x^5\)[/tex] by [tex]\(9y^4\)[/tex]:
[tex]\[ 2x^5 \cdot 9y^4 = 18x^5y^4 \][/tex]

3. Multiply [tex]\(3y^4\)[/tex] by [tex]\(-4x^2\)[/tex]:
[tex]\[ 3y^4 \cdot (-4x^2) = -12x^2y^4 \][/tex]

4. Multiply [tex]\(3y^4\)[/tex] by [tex]\(9y^4\)[/tex]:
[tex]\[ 3y^4 \cdot 9y^4 = 27y^{4+4} = 27y^8 \][/tex]

Now, let's sum up all these terms:
[tex]\[ -8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8 \][/tex]

Thus, the correct expression is:
[tex]\[ -8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8 \][/tex]

Comparing this with the provided options, the correct answer is:

C. [tex]\(-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8\)[/tex]

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