Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], follow these steps:
1. Simplify Each Component:
- We have three components to multiply:
[tex]\((7x^2)\)[/tex], [tex]\((2x^3 + 5)\)[/tex], and [tex]\((x^2 - 4x - 9)\)[/tex].
2. Multiply the First Two Factors:
- First, multiply [tex]\(7x^2\)[/tex] by the second factor, [tex]\((2x^3 + 5)\)[/tex].
- [tex]\((7x^2) \times (2x^3)\)[/tex] = [tex]\(14x^5\)[/tex].
- [tex]\((7x^2) \times 5\)[/tex] = [tex]\(35x^2\)[/tex].
- So, the result of this multiplication is [tex]\(14x^5 + 35x^2\)[/tex].
3. Multiply the Result with the Third Factor:
- Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by [tex]\((x^2 - 4x - 9)\)[/tex]:
- Multiply each term of [tex]\(14x^5 + 35x^2\)[/tex] by each term of the polynomial [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]
- [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]
4. Combine All Terms:
- Now, add together all these terms:
- [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex]
This gives us the expanded polynomial:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the final product of the expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. Simplify Each Component:
- We have three components to multiply:
[tex]\((7x^2)\)[/tex], [tex]\((2x^3 + 5)\)[/tex], and [tex]\((x^2 - 4x - 9)\)[/tex].
2. Multiply the First Two Factors:
- First, multiply [tex]\(7x^2\)[/tex] by the second factor, [tex]\((2x^3 + 5)\)[/tex].
- [tex]\((7x^2) \times (2x^3)\)[/tex] = [tex]\(14x^5\)[/tex].
- [tex]\((7x^2) \times 5\)[/tex] = [tex]\(35x^2\)[/tex].
- So, the result of this multiplication is [tex]\(14x^5 + 35x^2\)[/tex].
3. Multiply the Result with the Third Factor:
- Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by [tex]\((x^2 - 4x - 9)\)[/tex]:
- Multiply each term of [tex]\(14x^5 + 35x^2\)[/tex] by each term of the polynomial [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]
- [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]
4. Combine All Terms:
- Now, add together all these terms:
- [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex]
This gives us the expanded polynomial:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the final product of the expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
