Answer :
Sure! Let's multiply the two polynomials step by step:
We have two polynomials to multiply:
[tex]\[ (3x^2 - 4x + 5) \quad \text{and} \quad (x^2 - 3x + 2) \][/tex]
To multiply these polynomials, we'll use the distributive property, which means we'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\( 3x^2 \)[/tex] by each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
- [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]
- Multiply [tex]\(-4x\)[/tex] by each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- Multiply [tex]\(5\)[/tex] by each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all these results:
[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] term: [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]
Putting it all together, the multiplied polynomial is:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
So, the correct answer is:
[tex]\[
\boxed{3x^4 - 13x^3 + 23x^2 - 23x + 10}
\][/tex]
This matches option B from the choices given.
We have two polynomials to multiply:
[tex]\[ (3x^2 - 4x + 5) \quad \text{and} \quad (x^2 - 3x + 2) \][/tex]
To multiply these polynomials, we'll use the distributive property, which means we'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\( 3x^2 \)[/tex] by each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
- [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]
- Multiply [tex]\(-4x\)[/tex] by each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times -3x = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
- Multiply [tex]\(5\)[/tex] by each term in [tex]\( (x^2 - 3x + 2) \)[/tex]:
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times -3x = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
2. Combine all these results:
[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] term: [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]
Putting it all together, the multiplied polynomial is:
[tex]\[ 3x^4 - 13x^3 + 23x^2 - 23x + 10 \][/tex]
So, the correct answer is:
[tex]\[
\boxed{3x^4 - 13x^3 + 23x^2 - 23x + 10}
\][/tex]
This matches option B from the choices given.
