Answer :
To solve the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], follow these steps to find the product:
1. Distribute the First Term:
Start by distributing [tex]\(7x^2\)[/tex] to each term in the expression [tex]\(2x^3 + 5\)[/tex].
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the Result by the Third Polynomial:
Now, take the result from step 1, [tex]\(14x^5 + 35x^2\)[/tex], and distribute it to each term in [tex]\(x^2 - 4x - 9\)[/tex].
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
- First, distribute [tex]\(14x^5\)[/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]
- Next, distribute [tex]\(35x^2\)[/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]
3. Combine All Terms:
Combine all the terms you calculated above:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This results in the final product of the expression. The polynomial is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
1. Distribute the First Term:
Start by distributing [tex]\(7x^2\)[/tex] to each term in the expression [tex]\(2x^3 + 5\)[/tex].
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the Result by the Third Polynomial:
Now, take the result from step 1, [tex]\(14x^5 + 35x^2\)[/tex], and distribute it to each term in [tex]\(x^2 - 4x - 9\)[/tex].
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
- First, distribute [tex]\(14x^5\)[/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]
- Next, distribute [tex]\(35x^2\)[/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]
3. Combine All Terms:
Combine all the terms you calculated above:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This results in the final product of the expression. The polynomial is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]