Answer :
To find the product of the given expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], let's multiply the expressions step by step.
1. Multiply the first two terms: [tex]\(7x^2\)[/tex] and [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
- [tex]\(7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the result is [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third term [tex]\((x^2 - 4x - 9)\)[/tex]:
Now, take each term from [tex]\(14x^5 + 35x^2\)[/tex] and distribute it across [tex]\((x^2 - 4x - 9)\)[/tex].
- First, multiply [tex]\(14x^5\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \cdot (x^2 - 4x - 9) = 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
- [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \cdot (-9) = -126x^5\)[/tex]
- Next, multiply [tex]\(35x^2\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \cdot (x^2 - 4x - 9) = 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
- [tex]\(35x^2 \cdot x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \cdot (-9) = -315x^2\)[/tex]
3. Combine all the terms:
We sum all produced terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form of the product of the given expression.
1. Multiply the first two terms: [tex]\(7x^2\)[/tex] and [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
- [tex]\(7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the result is [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third term [tex]\((x^2 - 4x - 9)\)[/tex]:
Now, take each term from [tex]\(14x^5 + 35x^2\)[/tex] and distribute it across [tex]\((x^2 - 4x - 9)\)[/tex].
- First, multiply [tex]\(14x^5\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \cdot (x^2 - 4x - 9) = 14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\][/tex]
- [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \cdot (-9) = -126x^5\)[/tex]
- Next, multiply [tex]\(35x^2\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \cdot (x^2 - 4x - 9) = 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
- [tex]\(35x^2 \cdot x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \cdot (-9) = -315x^2\)[/tex]
3. Combine all the terms:
We sum all produced terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form of the product of the given expression.
