Answer :
To multiply the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we use the distributive property, often called the FOIL method when working with binomials. Let's expand the expression step by step:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\(3x^2 \times x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \times (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \times 2 = 6x^2\)[/tex]
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times (-3x) = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all the terms:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
3. Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(3x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-8x - 15x = -23x\)[/tex]
- Constant term: [tex]\(10\)[/tex]
4. Final expression:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
Therefore, the correct answer is:
D. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\(3x^2 \times x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \times (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \times 2 = 6x^2\)[/tex]
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times (-3x) = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all the terms:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
3. Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(3x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-8x - 15x = -23x\)[/tex]
- Constant term: [tex]\(10\)[/tex]
4. Final expression:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
Therefore, the correct answer is:
D. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]
