Answer :
Sure! Let's go through the steps to multiply the polynomials [tex]\((x+3)(3x^2+8x+9)\)[/tex].
1. Distribute [tex]\(x\)[/tex] across the second polynomial:
[tex]\[
x \cdot (3x^2 + 8x + 9) = 3x^3 + 8x^2 + 9x
\][/tex]
2. Distribute [tex]\(3\)[/tex] across the second polynomial:
[tex]\[
3 \cdot (3x^2 + 8x + 9) = 9x^2 + 24x + 27
\][/tex]
3. Add the results from steps 1 and 2:
Combine the like terms from the two results:
[tex]\[
(3x^3 + 8x^2 + 9x) + (9x^2 + 24x + 27)
\][/tex]
- [tex]\(3x^3\)[/tex] is alone, so it stays as [tex]\(3x^3\)[/tex].
- Add the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 9x^2 = 17x^2\)[/tex].
- Add the [tex]\(x\)[/tex] terms: [tex]\(9x + 24x = 33x\)[/tex].
- The constant term is [tex]\(27\)[/tex].
So, the product of the polynomials is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
The correct answer is C: [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
1. Distribute [tex]\(x\)[/tex] across the second polynomial:
[tex]\[
x \cdot (3x^2 + 8x + 9) = 3x^3 + 8x^2 + 9x
\][/tex]
2. Distribute [tex]\(3\)[/tex] across the second polynomial:
[tex]\[
3 \cdot (3x^2 + 8x + 9) = 9x^2 + 24x + 27
\][/tex]
3. Add the results from steps 1 and 2:
Combine the like terms from the two results:
[tex]\[
(3x^3 + 8x^2 + 9x) + (9x^2 + 24x + 27)
\][/tex]
- [tex]\(3x^3\)[/tex] is alone, so it stays as [tex]\(3x^3\)[/tex].
- Add the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 9x^2 = 17x^2\)[/tex].
- Add the [tex]\(x\)[/tex] terms: [tex]\(9x + 24x = 33x\)[/tex].
- The constant term is [tex]\(27\)[/tex].
So, the product of the polynomials is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
The correct answer is C: [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex].
