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Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], follow these steps:
1. Multiply the constants and the powers of [tex]\(x\)[/tex]:
Begin by multiplying the first term [tex]\(7x^2\)[/tex] with the polynomial [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. Multiply the result with the remaining polynomial:
Now, take the result from step 1, [tex]\((14x^5 + 35x^2)\)[/tex], and multiply it with [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Distribute both terms across each term in the second polynomial:
For [tex]\(14x^5\)[/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]
For [tex]\(35x^2\)[/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 the terms:
Add all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
4. Final Simplified Expression:
After combining, your final expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the product of the given expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
1. Multiply the constants and the powers of [tex]\(x\)[/tex]:
Begin by multiplying the first term [tex]\(7x^2\)[/tex] with the polynomial [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. Multiply the result with the remaining polynomial:
Now, take the result from step 1, [tex]\((14x^5 + 35x^2)\)[/tex], and multiply it with [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Distribute both terms across each term in the second polynomial:
For [tex]\(14x^5\)[/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]
For [tex]\(35x^2\)[/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 the terms:
Add all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
4. Final Simplified Expression:
After combining, your final expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the product of the given expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
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