Answer :
To find the product [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we can start by breaking down the process into simpler steps and multiplying each part step-by-step.
1. Step 1: Multiply the first two expressions:
Start with multiplying [tex]\((7x^2)\)[/tex] by [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]
2. Step 2: Multiply the result from Step 1 with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
Now, multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]. We will distribute each term in the first expression over the polynomial in the second expression.
- Multiply [tex]\(14x^5\)[/tex] by [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]
- Multiply [tex]\(35x^2\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/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:
Now, gather all the terms from the multiplications:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expanded expression is the product of the given expression. Thus, the result is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. Step 1: Multiply the first two expressions:
Start with multiplying [tex]\((7x^2)\)[/tex] by [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]
2. Step 2: Multiply the result from Step 1 with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
Now, multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]. We will distribute each term in the first expression over the polynomial in the second expression.
- Multiply [tex]\(14x^5\)[/tex] by [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]
- Multiply [tex]\(35x^2\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/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:
Now, gather all the terms from the multiplications:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expanded expression is the product of the given expression. Thus, the result is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
