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Select the correct answer.

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

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

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


Sagot :

To find the equivalent expression for the polynomial [tex]\(\left(2 x^5 + 3 y^4\right)\left(-4 x^2 + 9 y^4\right)\)[/tex], we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

Let's start by distributing each term:

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

Now, we'll perform each multiplication individually:

1. [tex]\((2x^5)(-4x^2) = -8x^{5+2} = -8x^7\)[/tex]
2. [tex]\((2x^5)(9y^4) = 18x^5y^4\)[/tex]
3. [tex]\((3y^4)(-4x^2) = -12x^2y^4\)[/tex]
4. [tex]\((3y^4)(9y^4) = 27y^{4+4} = 27y^8\)[/tex]

Now, combining these results:

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

This expression matches one of the options provided. Therefore, the correct answer is:

[tex]\[ \boxed{-8 x^7+18 x^5 y^4-12 x^2 y^4+27 y^8} \][/tex]

Hence, the correct answer is:
B. [tex]\(-8 x^7+18 x^5 y^4-12 x^2 y^4+27 y^8\)[/tex]