Answer :
Let's find the product of the given polynomials step by step:
We have three polynomials to multiply:
1. [tex]\( 7x^2 \)[/tex]
2. [tex]\( 2x^3 + 5 \)[/tex]
3. [tex]\( x^2 - 4x - 9 \)[/tex]
### Step 1: Multiply the first two polynomials
First, multiply [tex]\( 7x^2 \)[/tex] by [tex]\( 2x^3 + 5 \)[/tex]:
- Distribute [tex]\( 7x^2 \)[/tex] across each term in the polynomial [tex]\( 2x^3 + 5 \)[/tex]:
[tex]\[
\begin{align*}
(7x^2) \times (2x^3) &= 14x^5, \\
(7x^2) \times 5 &= 35x^2.
\end{align*}
\][/tex]
Combine these results:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third polynomial
Now, multiply [tex]\( 14x^5 + 35x^2 \)[/tex] by [tex]\( x^2 - 4x - 9 \)[/tex].
- Distribute each term of [tex]\( 14x^5 + 35x^2 \)[/tex] across [tex]\( x^2 - 4x - 9 \)[/tex]:
For [tex]\( 14x^5 \)[/tex]:
[tex]\[
\begin{align*}
(14x^5) \times (x^2) &= 14x^7, \\
(14x^5) \times (-4x) &= -56x^6, \\
(14x^5) \times (-9) &= -126x^5.
\end{align*}
\][/tex]
For [tex]\( 35x^2 \)[/tex]:
[tex]\[
\begin{align*}
(35x^2) \times (x^2) &= 35x^4, \\
(35x^2) \times (-4x) &= -140x^3, \\
(35x^2) \times (-9) &= -315x^2.
\end{align*}
\][/tex]
### Step 3: Combine all the terms
Now, combine all the like terms from the results above:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
That's the final product of the given multiplication:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
However, the given options did not correctly match our calculated polynomial, showing the results will not fit any of the provided options correctly. Make sure to double-check the problem statement and options.
We have three polynomials to multiply:
1. [tex]\( 7x^2 \)[/tex]
2. [tex]\( 2x^3 + 5 \)[/tex]
3. [tex]\( x^2 - 4x - 9 \)[/tex]
### Step 1: Multiply the first two polynomials
First, multiply [tex]\( 7x^2 \)[/tex] by [tex]\( 2x^3 + 5 \)[/tex]:
- Distribute [tex]\( 7x^2 \)[/tex] across each term in the polynomial [tex]\( 2x^3 + 5 \)[/tex]:
[tex]\[
\begin{align*}
(7x^2) \times (2x^3) &= 14x^5, \\
(7x^2) \times 5 &= 35x^2.
\end{align*}
\][/tex]
Combine these results:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third polynomial
Now, multiply [tex]\( 14x^5 + 35x^2 \)[/tex] by [tex]\( x^2 - 4x - 9 \)[/tex].
- Distribute each term of [tex]\( 14x^5 + 35x^2 \)[/tex] across [tex]\( x^2 - 4x - 9 \)[/tex]:
For [tex]\( 14x^5 \)[/tex]:
[tex]\[
\begin{align*}
(14x^5) \times (x^2) &= 14x^7, \\
(14x^5) \times (-4x) &= -56x^6, \\
(14x^5) \times (-9) &= -126x^5.
\end{align*}
\][/tex]
For [tex]\( 35x^2 \)[/tex]:
[tex]\[
\begin{align*}
(35x^2) \times (x^2) &= 35x^4, \\
(35x^2) \times (-4x) &= -140x^3, \\
(35x^2) \times (-9) &= -315x^2.
\end{align*}
\][/tex]
### Step 3: Combine all the terms
Now, combine all the like terms from the results above:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
That's the final product of the given multiplication:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
However, the given options did not correctly match our calculated polynomial, showing the results will not fit any of the provided options correctly. Make sure to double-check the problem statement and options.
