High School

Which of these is equivalent to the product of the expressions [tex]3x^2 - 7[/tex] and [tex]5x^3 - 8x - 6[/tex]?

A. [tex]8x^5 - 56x - 6[/tex]

B. [tex]5x^3 + 3x^2 - 8x - 13[/tex]

C. [tex]15x^5 - 59x^3 - 18x^2 + 56x + 42[/tex]

D. [tex]75x^6 - 35x^3 - 42x^2 + 56x + 42[/tex]

Answer :

We want to find the product of the two polynomials

[tex]$$3x^2 - 7$$[/tex]

and

[tex]$$5x^3 - 8x - 6.$$[/tex]

Step 1: Multiply each term in the first polynomial by each term in the second polynomial.

- Multiply [tex]\(3x^2\)[/tex] by each term in [tex]\(5x^3 - 8x - 6\)[/tex]:

[tex]\[
\begin{aligned}
3x^2 \cdot 5x^3 &= 15x^5, \\
3x^2 \cdot (-8x) &= -24x^3, \\
3x^2 \cdot (-6) &= -18x^2.
\end{aligned}
\][/tex]

- Multiply [tex]\(-7\)[/tex] by each term in [tex]\(5x^3 - 8x - 6\)[/tex]:

[tex]\[
\begin{aligned}
-7 \cdot 5x^3 &= -35x^3, \\
-7 \cdot (-8x) &= 56x, \\
-7 \cdot (-6) &= 42.
\end{aligned}
\][/tex]

Step 2: Combine like terms.

List all the products we obtained:

- [tex]\(15x^5\)[/tex],
- [tex]\(-24x^3\)[/tex],
- [tex]\(-18x^2\)[/tex],
- [tex]\(-35x^3\)[/tex],
- [tex]\(56x\)[/tex],
- [tex]\(42\)[/tex].

Now, combine the like terms (in this case, the [tex]\(x^3\)[/tex] terms):

[tex]\[
-24x^3 - 35x^3 = -59x^3.
\][/tex]

So the entire expanded product is:

[tex]\[
15x^5 - 59x^3 - 18x^2 + 56x + 42.
\][/tex]

Step 3: Match with the provided options.

The expanded result

[tex]$$15x^5 - 59x^3 - 18x^2 + 56x + 42$$[/tex]

corresponds to option C.

Therefore, the final answer is option C.

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