Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will follow a detailed step-by-step process.
1. Multiply the First Two Parts:
Start with the expression [tex]\((7x^2)(2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] across each term inside the parentheses:
[tex]\[
7x^2 \times 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \times 5 = 35x^2
\][/tex]
So, the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Now Multiply the Result with the Third Part:
Now we will multiply the intermediate result [tex]\((14x^5 + 35x^2)\)[/tex] with the third part [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 in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
- Multiply [tex]\(35x^2\)[/tex] with each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
3. Combine All the Terms:
Now, add all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This gives us the final expanded product of the expression.
The resulting polynomial is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This is the answer to the original question.
1. Multiply the First Two Parts:
Start with the expression [tex]\((7x^2)(2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] across each term inside the parentheses:
[tex]\[
7x^2 \times 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \times 5 = 35x^2
\][/tex]
So, the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Now Multiply the Result with the Third Part:
Now we will multiply the intermediate result [tex]\((14x^5 + 35x^2)\)[/tex] with the third part [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 in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \times (-9) = -126x^5
\][/tex]
- Multiply [tex]\(35x^2\)[/tex] with each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \times (-9) = -315x^2
\][/tex]
3. Combine All the Terms:
Now, add all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This gives us the final expanded product of the expression.
The resulting polynomial is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This is the answer to the original question.
