Answer :
To solve for the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will break it down into manageable steps:
1. Distribute the First Two Factors:
We start by distributing the first two terms: [tex]\((7x^2)(2x^3 + 5)\)[/tex].
- First, distribute [tex]\(7x^2\)[/tex] to [tex]\(2x^3\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
- Next, distribute [tex]\(7x^2\)[/tex] to [tex]\(5\)[/tex]:
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
This means our intermediate expression is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Distribute the Result with the Third Factor:
Next, we need to multiply the result by the third factor: [tex]\((x^2 - 4x - 9)\)[/tex].
Let's distribute each term in the binomial [tex]\((14x^5 + 35x^2)\)[/tex] through [tex]\((x^2 - 4x - 9)\)[/tex]:
- Distribute [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]
- Distribute [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:
Now, combine all the like terms from these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
That's the expanded polynomial representing the product of the original expression. The coefficients and the degrees of the terms match perfectly with what we found, confirming our solution.
1. Distribute the First Two Factors:
We start by distributing the first two terms: [tex]\((7x^2)(2x^3 + 5)\)[/tex].
- First, distribute [tex]\(7x^2\)[/tex] to [tex]\(2x^3\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
- Next, distribute [tex]\(7x^2\)[/tex] to [tex]\(5\)[/tex]:
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
This means our intermediate expression is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Distribute the Result with the Third Factor:
Next, we need to multiply the result by the third factor: [tex]\((x^2 - 4x - 9)\)[/tex].
Let's distribute each term in the binomial [tex]\((14x^5 + 35x^2)\)[/tex] through [tex]\((x^2 - 4x - 9)\)[/tex]:
- Distribute [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]
- Distribute [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:
Now, combine all the like terms from these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
That's the expanded polynomial representing the product of the original expression. The coefficients and the degrees of the terms match perfectly with what we found, confirming our solution.
