Answer :
Sure, let's go through the process step-by-step to find the product of the polynomials:
We need to compute the product of the following three polynomials:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)\left(x^2 - 4x - 9\right)
\][/tex]
### Step 1: Multiply the First Two Polynomials
Start with the first two polynomials:
[tex]\[
(7x^2) \cdot (2x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] inside the parenthesis:
[tex]\[
7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the Result with the Third Polynomial
Now, take the result [tex]\((14x^5 + 35x^2)\)[/tex] and multiply it by [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Expand it term-by-term:
[tex]\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) + 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
Calculate each term:
[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]
### Step 3: Combine All the Terms
Now, combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the final expanded product is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
We need to compute the product of the following three polynomials:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)\left(x^2 - 4x - 9\right)
\][/tex]
### Step 1: Multiply the First Two Polynomials
Start with the first two polynomials:
[tex]\[
(7x^2) \cdot (2x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] inside the parenthesis:
[tex]\[
7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the Result with the Third Polynomial
Now, take the result [tex]\((14x^5 + 35x^2)\)[/tex] and multiply it by [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Expand it term-by-term:
[tex]\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) + 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
Calculate each term:
[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]
### Step 3: Combine All the Terms
Now, combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the final expanded product is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
