Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will expand the expression step by step.
1. Distribute the first two factors: Start by multiplying the polynomial [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]
[tex]\[
= 14x^5 + 35x^2
\][/tex]
Now, we have our intermediate result: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third factor: Next, take the polynomial [tex]\(14x^5 + 35x^2\)[/tex] and multiply it with [tex]\((x^2 - 4x - 9)\)[/tex].
- First, multiply [tex]\(14x^5\)[/tex] by each term in the second polynomial:
[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]
- Then, multiply [tex]\(35x^2\)[/tex] by each term in the second polynomial:
[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 the terms: Finally, add all these products together to get the final expanded polynomial.
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. Distribute the first two factors: Start by multiplying the polynomial [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]
[tex]\[
= 14x^5 + 35x^2
\][/tex]
Now, we have our intermediate result: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third factor: Next, take the polynomial [tex]\(14x^5 + 35x^2\)[/tex] and multiply it with [tex]\((x^2 - 4x - 9)\)[/tex].
- First, multiply [tex]\(14x^5\)[/tex] by each term in the second polynomial:
[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]
- Then, multiply [tex]\(35x^2\)[/tex] by each term in the second polynomial:
[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 the terms: Finally, add all these products together to get the final expanded polynomial.
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
