Answer :
To multiply the polynomials
[tex]$$
(7x^2 + 9x + 7)(9x - 4),
$$[/tex]
we apply the distributive property (also known as the FOIL method for binomials). Each term in the first polynomial is multiplied by each term in the second polynomial:
1. Multiply [tex]$7x^2$[/tex] by each term in [tex]$9x - 4$[/tex]:
[tex]\[
7x^2 \cdot 9x = 63x^3 \quad \text{and} \quad 7x^2 \cdot (-4) = -28x^2.
\][/tex]
2. Multiply [tex]$9x$[/tex] by each term in [tex]$9x - 4$[/tex]:
[tex]\[
9x \cdot 9x = 81x^2 \quad \text{and} \quad 9x \cdot (-4) = -36x.
\][/tex]
3. Multiply [tex]$7$[/tex] by each term in [tex]$9x - 4$[/tex]:
[tex]\[
7 \cdot 9x = 63x \quad \text{and} \quad 7 \cdot (-4) = -28.
\][/tex]
Now, combine the like terms:
- For the [tex]$x^3$[/tex] term:
[tex]\[
63x^3.
\][/tex]
- For the [tex]$x^2$[/tex] terms:
[tex]\[
-28x^2 + 81x^2 = 53x^2.
\][/tex]
- For the [tex]$x$[/tex] terms:
[tex]\[
-36x + 63x = 27x.
\][/tex]
- The constant term is:
[tex]\[
-28.
\][/tex]
Thus, the expanded form of the polynomial is:
[tex]$$
63x^3 + 53x^2 + 27x - 28.
$$[/tex]
Comparing with the provided options, we see that the correct answer is:
Option B.
[tex]$$
(7x^2 + 9x + 7)(9x - 4),
$$[/tex]
we apply the distributive property (also known as the FOIL method for binomials). Each term in the first polynomial is multiplied by each term in the second polynomial:
1. Multiply [tex]$7x^2$[/tex] by each term in [tex]$9x - 4$[/tex]:
[tex]\[
7x^2 \cdot 9x = 63x^3 \quad \text{and} \quad 7x^2 \cdot (-4) = -28x^2.
\][/tex]
2. Multiply [tex]$9x$[/tex] by each term in [tex]$9x - 4$[/tex]:
[tex]\[
9x \cdot 9x = 81x^2 \quad \text{and} \quad 9x \cdot (-4) = -36x.
\][/tex]
3. Multiply [tex]$7$[/tex] by each term in [tex]$9x - 4$[/tex]:
[tex]\[
7 \cdot 9x = 63x \quad \text{and} \quad 7 \cdot (-4) = -28.
\][/tex]
Now, combine the like terms:
- For the [tex]$x^3$[/tex] term:
[tex]\[
63x^3.
\][/tex]
- For the [tex]$x^2$[/tex] terms:
[tex]\[
-28x^2 + 81x^2 = 53x^2.
\][/tex]
- For the [tex]$x$[/tex] terms:
[tex]\[
-36x + 63x = 27x.
\][/tex]
- The constant term is:
[tex]\[
-28.
\][/tex]
Thus, the expanded form of the polynomial is:
[tex]$$
63x^3 + 53x^2 + 27x - 28.
$$[/tex]
Comparing with the provided options, we see that the correct answer is:
Option B.
