Answer :
To find the product of the given expression [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], let's break it down into manageable steps.
1. Distribute the First Polynomial:
- Start by multiplying [tex]\(7x^2\)[/tex] with each term of the second polynomial [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2) \times (2x^3 + 5) = 14x^5 + 35x^2
\][/tex]
2. Expand the Product:
- Now, take the result from step 1 and multiply it with the third polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
- Distribute each term of [tex]\(14x^5 + 35x^2\)[/tex] across the polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
3. Distribute Each Term:
- Distribute [tex]\(14x^5\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
- Distribute [tex]\(35x^2\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
4. Combine Like Terms:
- Sum up all the terms obtained from the distributions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is how you arrive at the final expanded product:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This expression represents the fully expanded form of the product of the given polynomials.
1. Distribute the First Polynomial:
- Start by multiplying [tex]\(7x^2\)[/tex] with each term of the second polynomial [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2) \times (2x^3 + 5) = 14x^5 + 35x^2
\][/tex]
2. Expand the Product:
- Now, take the result from step 1 and multiply it with the third polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
- Distribute each term of [tex]\(14x^5 + 35x^2\)[/tex] across the polynomial [tex]\((x^2 - 4x - 9)\)[/tex]:
3. Distribute Each Term:
- Distribute [tex]\(14x^5\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
- Distribute [tex]\(35x^2\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
4. Combine Like Terms:
- Sum up all the terms obtained from the distributions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is how you arrive at the final expanded product:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This expression represents the fully expanded form of the product of the given polynomials.
