Answer :
To find the product of the expression [tex]\(\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)\)[/tex], we'll work through the calculation in a step-by-step manner.
1. First, distribute [tex]\((7x^2)\)[/tex] to the terms in [tex]\((2x^3 + 5)\)[/tex]:
Multiply [tex]\(7x^2\)[/tex] by each term in [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2) \cdot (2x^3) = 14x^5
\][/tex]
[tex]\[
(7x^2) \cdot 5 = 35x^2
\][/tex]
So, we get: [tex]\(14x^5 + 35x^2\)[/tex].
2. Next, multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]:
We need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]. Distribute each term from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(14x^5\)[/tex] with each term:
[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] with each term:
[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 from the multiplication:
The expanded product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the resultant polynomial from multiplying the given expressions.
1. First, distribute [tex]\((7x^2)\)[/tex] to the terms in [tex]\((2x^3 + 5)\)[/tex]:
Multiply [tex]\(7x^2\)[/tex] by each term in [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2) \cdot (2x^3) = 14x^5
\][/tex]
[tex]\[
(7x^2) \cdot 5 = 35x^2
\][/tex]
So, we get: [tex]\(14x^5 + 35x^2\)[/tex].
2. Next, multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]:
We need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]. Distribute each term from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(14x^5\)[/tex] with each term:
[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] with each term:
[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 from the multiplication:
The expanded product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the resultant polynomial from multiplying the given expressions.
