To find the product of the two given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], follow these steps:
1. Identify the given functions:
[tex]\[
f(x) = 11x^3 - 3x^2
\][/tex]
[tex]\[
g(x) = 7x^4 + 9x^3
\][/tex]
2. Multiply [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
When you multiply two polynomials, you distribute each term in the first polynomial by each term in the second polynomial.
Let’s distribute each term:
[tex]\[
(11x^3 - 3x^2) \cdot (7x^4 + 9x^3)
\][/tex]
- Multiply [tex]\( 11x^3 \)[/tex] by each term in [tex]\( (7x^4 + 9x^3) \)[/tex]:
[tex]\[
11x^3 \cdot 7x^4 = 77x^7
\][/tex]
[tex]\[
11x^3 \cdot 9x^3 = 99x^6
\][/tex]
- Multiply [tex]\( -3x^2 \)[/tex] by each term in [tex]\( (7x^4 + 9x^3) \)[/tex]:
[tex]\[
-3x^2 \cdot 7x^4 = -21x^6
\][/tex]
[tex]\[
-3x^2 \cdot 9x^3 = -27x^5
\][/tex]
3. Combine like terms:
Now, combine all the products you have calculated:
[tex]\[
77x^7 + 99x^6 - 21x^6 - 27x^5
\][/tex]
Simplify the like terms:
[tex]\[
77x^7 + (99x^6 - 21x^6) - 27x^5
\][/tex]
[tex]\[
77x^7 + 78x^6 - 27x^5
\][/tex]
4. Compare to the given options:
Look at the choices provided to find the matching expression:
A. [tex]\( 7x^4 + mx^3 - 3x^2 \)[/tex]
B. [tex]\( \pi x^{12} + 99x^3 - 21x^4 - 27x^4 \)[/tex]
C. [tex]\( 18x^7 + 10x^4 + 6x^5 \)[/tex]
D. [tex]\( 77x^7 + 78x^6 - 27x^5 \)[/tex]
The correct expression that matches the product [tex]\( 77x^7 + 78x^6 - 27x^5 \)[/tex] is:
[tex]\[
\boxed{D}
\][/tex]
