Answer :
Sure, let's solve the given problem step-by-step to find the product of the polynomials:
The expression to simplify is:
[tex]\[ \left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right) \][/tex]
### Step 1: Multiply the first two polynomials
First, we will multiply [tex]\( 7x^2 \)[/tex] with [tex]\( 2x^3 + 5 \)[/tex].
[tex]\[
7x^2 \cdot (2x^3 + 5) = (7x^2) \cdot 2x^3 + (7x^2) \cdot 5
\][/tex]
This gives us:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
So, when we combine these, we get:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the resulting polynomial with the third polynomial
Next, we multiply [tex]\( 14x^5 + 35x^2 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex].
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
We will distribute [tex]\(14x^5\)[/tex] and [tex]\(35x^2\)[/tex] to each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]
### Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the final product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So the correct answer is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
The expression to simplify is:
[tex]\[ \left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right) \][/tex]
### Step 1: Multiply the first two polynomials
First, we will multiply [tex]\( 7x^2 \)[/tex] with [tex]\( 2x^3 + 5 \)[/tex].
[tex]\[
7x^2 \cdot (2x^3 + 5) = (7x^2) \cdot 2x^3 + (7x^2) \cdot 5
\][/tex]
This gives us:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
So, when we combine these, we get:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the resulting polynomial with the third polynomial
Next, we multiply [tex]\( 14x^5 + 35x^2 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex].
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
We will distribute [tex]\(14x^5\)[/tex] and [tex]\(35x^2\)[/tex] to each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]
### Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the final product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So the correct answer is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
