High School

Multiply the polynomials:

[tex]
\left(7x^2 + 9x + 7\right)(9x - 4)
[/tex]

A. [tex]63x^3 + 53x^2 + 27x + 28[/tex]
B. [tex]63x^3 + 53x^2 + 59x - 28[/tex]
C. [tex]63x^3 + 81x^2 + 27x - 28[/tex]
D. [tex]63x^3 + 53x^2 + 27x - 28[/tex]

Answer :

Sure! To multiply the polynomials [tex]\((7x^2 + 9x + 7)(9x - 4)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Here's a step-by-step process:

1. Distribute each term of the first polynomial to each term of the second polynomial:

[tex]\[
(7x^2 + 9x + 7)(9x - 4) = 7x^2(9x - 4) + 9x(9x - 4) + 7(9x - 4)
\][/tex]

2. Multiply each term inside the parentheses:

[tex]\[
7x^2(9x) + 7x^2(-4) + 9x(9x) + 9x(-4) + 7(9x) + 7(-4)
\][/tex]

3. Calculate each product:

- [tex]\(7x^2 \cdot 9x = 63x^3\)[/tex]
- [tex]\(7x^2 \cdot -4 = -28x^2\)[/tex]
- [tex]\(9x \cdot 9x = 81x^2\)[/tex]
- [tex]\(9x \cdot -4 = -36x\)[/tex]
- [tex]\(7 \cdot 9x = 63x\)[/tex]
- [tex]\(7 \cdot -4 = -28\)[/tex]

4. Combine all the terms:

[tex]\[
63x^3 - 28x^2 + 81x^2 - 36x + 63x - 28
\][/tex]

5. Combine like terms:

- Combine the [tex]\(x^2\)[/tex] terms:

[tex]\[
-28x^2 + 81x^2 = 53x^2
\][/tex]

- Combine the [tex]\(x\)[/tex] terms:

[tex]\[
-36x + 63x = 27x
\][/tex]

So, the resulting polynomial is:

[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]

Therefore, the correct answer is:

D. [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex]

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