College

Multiply the following expression:

[tex]\left(3x^2 - 4x + 5\right)\left(x^2 - 3x + 2\right)[/tex]

A. [tex]3x^4 + 10x^2 + 12x + 10[/tex]

B. [tex]4x^2 - 7x + 7[/tex]

C. [tex]3x^4 - 13x^3 + 23x^2 - 23x + 10[/tex]

D. [tex]3x^4 + 12x^2 + 10[/tex]

Answer :

To multiply the two polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we will expand the expression by distributing each term in the first polynomial to each term in the second polynomial.

Here’s how you do it step-by-step:

1. Multiply each term of [tex]\((3x^2 - 4x + 5)\)[/tex] by every term in [tex]\((x^2 - 3x + 2)\)[/tex]:

- First term: [tex]\(3x^2\)[/tex]:
- [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]

- Second term: [tex]\(-4x\)[/tex]:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]

- Third term: [tex]\(5\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]

2. Combine all the results:

- Collect and write down all terms:
[tex]\[3x^4, -9x^3, 6x^2, -4x^3, 12x^2, -8x, 5x^2, -15x, 10\][/tex]

3. Combine like terms:

- The [tex]\(x^4\)[/tex] term: [tex]\(3x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- The constant term: [tex]\(10\)[/tex]

4. Write down the final result:

[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]

Therefore, the correct answer is C. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].