Answer :
- Simplify $f(x)$ to $8x^3$.
- Multiply $f(x)$ and $g(x)$: $(8x^3)(7x^4 + 9x^3)$.
- Distribute and simplify: $56x^7 + 72x^6$.
- Compare the result with the options. There appears to be a typo in the provided options, as none match the correct expression. The correct answer should be $56x^7 + 72x^6$.
### Explanation
1. Simplify f(x)
First, we need to simplify the expression for $f(x)$. We have $f(x) = 11x^3 - 3x^3$. Combining like terms, we get $f(x) = (11-3)x^3 = 8x^3$.
2. Multiply f(x) and g(x)
Next, we need to find the product of $f(x)$ and $g(x)$. We have $f(x) = 8x^3$ and $g(x) = 7x^4 + 9x^3$. So, $f(x) cdot g(x) = (8x^3)(7x^4 + 9x^3)$.
3. Distribute and Simplify
Now, we distribute $8x^3$ to both terms in $g(x)$:
$f(x) cdot g(x) = 8x^3 cdot 7x^4 + 8x^3 cdot 9x^3$
$f(x) cdot g(x) = 56x^{3+4} + 72x^{3+3}$
$f(x) cdot g(x) = 56x^7 + 72x^6$
4. Compare with Options
Finally, we compare our result $56x^7 + 72x^6$ with the given options:
A. $77 x^7+78 x^6-27 x^6 = 77x^7 + 51x^6$
B. $77 x^{13}+99 x^9-21 x^8-27 x^6$
C. $7 x^4+99 x^3-3 x^2$
D. $18 x^7+10 x^6+6 x^8$
None of the options exactly match our result. However, let's re-examine the problem statement and our calculations to ensure we didn't make a mistake. We have $f(x) = 11x^3 - 3x^3 = 8x^3$ and $g(x) = 7x^4 + 9x^3$. Then, $f(x) cdot g(x) = (8x^3)(7x^4 + 9x^3) = 56x^7 + 72x^6$. It seems there might be a typo in the options provided. However, if we look closely at option A, $77 x^7+78 x^6-27 x^6 = 77x^7 + (78-27)x^6 = 77x^7 + 51x^6$, which is still not the correct answer. There seems to be an error in the provided options. The correct expression should be $56x^7 + 72x^6$.
### Examples
Understanding polynomial multiplication is crucial in various fields like engineering, computer graphics, and physics. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution. Multiplying these polynomials helps them predict the overall stress and ensure the bridge's stability under different conditions. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces, enhancing the visual appeal and realism of digital models.
- Multiply $f(x)$ and $g(x)$: $(8x^3)(7x^4 + 9x^3)$.
- Distribute and simplify: $56x^7 + 72x^6$.
- Compare the result with the options. There appears to be a typo in the provided options, as none match the correct expression. The correct answer should be $56x^7 + 72x^6$.
### Explanation
1. Simplify f(x)
First, we need to simplify the expression for $f(x)$. We have $f(x) = 11x^3 - 3x^3$. Combining like terms, we get $f(x) = (11-3)x^3 = 8x^3$.
2. Multiply f(x) and g(x)
Next, we need to find the product of $f(x)$ and $g(x)$. We have $f(x) = 8x^3$ and $g(x) = 7x^4 + 9x^3$. So, $f(x) cdot g(x) = (8x^3)(7x^4 + 9x^3)$.
3. Distribute and Simplify
Now, we distribute $8x^3$ to both terms in $g(x)$:
$f(x) cdot g(x) = 8x^3 cdot 7x^4 + 8x^3 cdot 9x^3$
$f(x) cdot g(x) = 56x^{3+4} + 72x^{3+3}$
$f(x) cdot g(x) = 56x^7 + 72x^6$
4. Compare with Options
Finally, we compare our result $56x^7 + 72x^6$ with the given options:
A. $77 x^7+78 x^6-27 x^6 = 77x^7 + 51x^6$
B. $77 x^{13}+99 x^9-21 x^8-27 x^6$
C. $7 x^4+99 x^3-3 x^2$
D. $18 x^7+10 x^6+6 x^8$
None of the options exactly match our result. However, let's re-examine the problem statement and our calculations to ensure we didn't make a mistake. We have $f(x) = 11x^3 - 3x^3 = 8x^3$ and $g(x) = 7x^4 + 9x^3$. Then, $f(x) cdot g(x) = (8x^3)(7x^4 + 9x^3) = 56x^7 + 72x^6$. It seems there might be a typo in the options provided. However, if we look closely at option A, $77 x^7+78 x^6-27 x^6 = 77x^7 + (78-27)x^6 = 77x^7 + 51x^6$, which is still not the correct answer. There seems to be an error in the provided options. The correct expression should be $56x^7 + 72x^6$.
### Examples
Understanding polynomial multiplication is crucial in various fields like engineering, computer graphics, and physics. For instance, when designing a bridge, engineers use polynomial functions to model the load distribution. Multiplying these polynomials helps them predict the overall stress and ensure the bridge's stability under different conditions. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces, enhancing the visual appeal and realism of digital models.
