Answer :
To find the product of the given expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we can follow these steps:
1. Start by expanding the expressions:
- Take [tex]\((7x^2)\)[/tex] and multiply it with each term inside the parenthesis for the other expressions.
2. Multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:
- Distribute [tex]\(7x^2\)[/tex] across each term in the binomial:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the product is: [tex]\[14x^5 + 35x^2\][/tex]
3. Multiply the result by the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
- Now distribute each term of [tex]\((14x^5 + 35x^2)\)[/tex] across each term in [tex]\((x^2 - 4x - 9)\)[/tex].
Here’s how you do it:
- 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]
4. Combine all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
5. Final Expression:
- This is the expanded form of the product of the given expressions.
The final expression is [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex], which matches the third option in the original multiple-choice question.
1. Start by expanding the expressions:
- Take [tex]\((7x^2)\)[/tex] and multiply it with each term inside the parenthesis for the other expressions.
2. Multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:
- Distribute [tex]\(7x^2\)[/tex] across each term in the binomial:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the product is: [tex]\[14x^5 + 35x^2\][/tex]
3. Multiply the result by the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
- Now distribute each term of [tex]\((14x^5 + 35x^2)\)[/tex] across each term in [tex]\((x^2 - 4x - 9)\)[/tex].
Here’s how you do it:
- 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]
4. Combine all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
5. Final Expression:
- This is the expanded form of the product of the given expressions.
The final expression is [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex], which matches the third option in the original multiple-choice question.
