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Answer :
Sure! Let's solve the problem by multiplying the expressions step-by-step:
We need to find the product of [tex]\((7x^2)(2x^3+5)(x^2-4x-9)\)[/tex].
### Step 1: Expand the first pair of terms
First, multiply [tex]\(7x^2\)[/tex] with the polynomial [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculate each term:
- [tex]\(7x^2 \cdot 2x^3 = 14x^{5}\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, [tex]\((7x^2)(2x^3 + 5) = 14x^5 + 35x^2\)[/tex].
### Step 2: Multiply the result with the remaining polynomial
Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it with [tex]\(x^2 - 4x - 9\)[/tex]:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
Distribute each term from the first polynomial through 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]
- [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]
### Step 3: Combine like terms
Combine all the terms we found:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded and simplified product of the original expression.
Therefore, the final product is:
[tex]\[ \boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2} \][/tex]
We need to find the product of [tex]\((7x^2)(2x^3+5)(x^2-4x-9)\)[/tex].
### Step 1: Expand the first pair of terms
First, multiply [tex]\(7x^2\)[/tex] with the polynomial [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculate each term:
- [tex]\(7x^2 \cdot 2x^3 = 14x^{5}\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, [tex]\((7x^2)(2x^3 + 5) = 14x^5 + 35x^2\)[/tex].
### Step 2: Multiply the result with the remaining polynomial
Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it with [tex]\(x^2 - 4x - 9\)[/tex]:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
Distribute each term from the first polynomial through 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]
- [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]
### Step 3: Combine like terms
Combine all the terms we found:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded and simplified product of the original expression.
Therefore, the final product is:
[tex]\[ \boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2} \][/tex]
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