Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will go through the step-by-step multiplication of these polynomials. Here’s how you can do it:
1. Distribute the First Part: Start by distributing [tex]\(7x^2\)[/tex] into the first polynomial [tex]\((2x^3 + 5)\)[/tex].
- Multiply [tex]\(7x^2 \times 2x^3\)[/tex] to get [tex]\(14x^5\)[/tex].
- Multiply [tex]\(7x^2 \times 5\)[/tex] to get [tex]\(35x^2\)[/tex].
So, the product of the first two parts is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply by the Second Polynomial: Now, take the result from the first step and distribute it into the second polynomial [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5 \times x^2\)[/tex] to get [tex]\(14x^7\)[/tex].
- Multiply [tex]\(14x^5 \times (-4x)\)[/tex] to get [tex]\(-56x^6\)[/tex].
- Multiply [tex]\(14x^5 \times (-9)\)[/tex] to get [tex]\(-126x^5\)[/tex].
- Multiply [tex]\(35x^2 \times x^2\)[/tex] to get [tex]\(35x^4\)[/tex].
- Multiply [tex]\(35x^2 \times (-4x)\)[/tex] to get [tex]\(-140x^3\)[/tex].
- Multiply [tex]\(35x^2 \times (-9)\)[/tex] to get [tex]\(-315x^2\)[/tex].
Combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
3. Final Answer: After combining and simplifying all the terms, the complete expanded product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So the product of the given polynomials is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
1. Distribute the First Part: Start by distributing [tex]\(7x^2\)[/tex] into the first polynomial [tex]\((2x^3 + 5)\)[/tex].
- Multiply [tex]\(7x^2 \times 2x^3\)[/tex] to get [tex]\(14x^5\)[/tex].
- Multiply [tex]\(7x^2 \times 5\)[/tex] to get [tex]\(35x^2\)[/tex].
So, the product of the first two parts is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply by the Second Polynomial: Now, take the result from the first step and distribute it into the second polynomial [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5 \times x^2\)[/tex] to get [tex]\(14x^7\)[/tex].
- Multiply [tex]\(14x^5 \times (-4x)\)[/tex] to get [tex]\(-56x^6\)[/tex].
- Multiply [tex]\(14x^5 \times (-9)\)[/tex] to get [tex]\(-126x^5\)[/tex].
- Multiply [tex]\(35x^2 \times x^2\)[/tex] to get [tex]\(35x^4\)[/tex].
- Multiply [tex]\(35x^2 \times (-4x)\)[/tex] to get [tex]\(-140x^3\)[/tex].
- Multiply [tex]\(35x^2 \times (-9)\)[/tex] to get [tex]\(-315x^2\)[/tex].
Combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
3. Final Answer: After combining and simplifying all the terms, the complete expanded product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So the product of the given polynomials is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
