College

Multiply the polynomials:

\[ (x+3)\left(3x^2+8x+9\right) \]

A. \[ 3x^3 + 17x^2 + 33x - 27 \]

B. \[ 3x^3 + 17x^2 + 33x + 27 \]

C. \[ 3x^3 + x^2 + 33x + 27 \]

D. \[ 3x^3 + 17x^2 - 15x + 27 \]

Answer :

To multiply the polynomials [tex]\((x + 3)(3x^2 + 8x + 9)\)[/tex], follow these steps:

1. Distribute each term in the first polynomial with each term in the second polynomial:

- Multiply [tex]\(x\)[/tex] by each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
- [tex]\(x \cdot 3x^2 = 3x^3\)[/tex]
- [tex]\(x \cdot 8x = 8x^2\)[/tex]
- [tex]\(x \cdot 9 = 9x\)[/tex]

- Multiply [tex]\(3\)[/tex] by each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
- [tex]\(3 \cdot 3x^2 = 9x^2\)[/tex]
- [tex]\(3 \cdot 8x = 24x\)[/tex]
- [tex]\(3 \cdot 9 = 27\)[/tex]

2. Combine all the results:
[tex]\[
3x^3 + 8x^2 + 9x + 9x^2 + 24x + 27
\][/tex]

3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 9x^2 = 17x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(9x + 24x = 33x\)[/tex]

4. Write down the simplified polynomial:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]

The result is [tex]\(3x^3 + 17x^2 + 33x + 27\)[/tex], which matches option B.