Answer :
To solve the multiplication of the expressions [tex]\((9x + 8)\)[/tex] and [tex]\((3x^2 + x - 1)\)[/tex], we can use the distributive property, also known as the FOIL method for binomials. However, since one expression is a binomial and the other is a trinomial, we'll distribute each term in the first expression across every term in the second expression. Let's break it down step-by-step:
1. Distribute [tex]\(9x\)[/tex]:
- Multiply [tex]\(9x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
9x \cdot 3x^2 = 27x^3
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
9x \cdot x = 9x^2
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[
9x \cdot -1 = -9x
\][/tex]
2. Distribute [tex]\(8\)[/tex]:
- Multiply [tex]\(8\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
8 \cdot 3x^2 = 24x^2
\][/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
8 \cdot x = 8x
\][/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[
8 \cdot -1 = -8
\][/tex]
3. Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(27x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(9x^2 + 24x^2\)[/tex]:
[tex]\[
9x^2 + 24x^2 = 33x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-9x + 8x\)[/tex]:
[tex]\[
-9x + 8x = -1x
\][/tex]
- The constant term is [tex]\(-8\)[/tex].
Putting it all together, the expanded polynomial is:
[tex]\[
27x^3 + 33x^2 - x - 8
\][/tex]
This is the result of multiplying the two expressions.
1. Distribute [tex]\(9x\)[/tex]:
- Multiply [tex]\(9x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
9x \cdot 3x^2 = 27x^3
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
9x \cdot x = 9x^2
\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[
9x \cdot -1 = -9x
\][/tex]
2. Distribute [tex]\(8\)[/tex]:
- Multiply [tex]\(8\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[
8 \cdot 3x^2 = 24x^2
\][/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
8 \cdot x = 8x
\][/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[
8 \cdot -1 = -8
\][/tex]
3. Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(27x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(9x^2 + 24x^2\)[/tex]:
[tex]\[
9x^2 + 24x^2 = 33x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-9x + 8x\)[/tex]:
[tex]\[
-9x + 8x = -1x
\][/tex]
- The constant term is [tex]\(-8\)[/tex].
Putting it all together, the expanded polynomial is:
[tex]\[
27x^3 + 33x^2 - x - 8
\][/tex]
This is the result of multiplying the two expressions.
