Answer :
To multiply the polynomials [tex]\( (3x^2 - 4x + 5) \)[/tex] and [tex]\( (x^2 - 3x + 2) \)[/tex], we'll use the distributive property and carefully combine like terms. Here's a step-by-step guide:
1. Distribute each term from the first polynomial across each term from the second polynomial:
- First term of [tex]\(3x^2\)[/tex] with all terms of the second polynomial:
[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 of [tex]\(-4x\)[/tex] with all terms of the second polynomial:
[tex]\[
-4x \cdot x^2 = -4x^3
\][/tex]
[tex]\[
-4x \cdot (-3x) = 12x^2
\][/tex]
[tex]\[
-4x \cdot 2 = -8x
\][/tex]
- Third term of [tex]\(5\)[/tex] with all terms of the second polynomial:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-3x) = -15x
\][/tex]
[tex]\[
5 \cdot 2 = 10
\][/tex]
2. Combine all the distributed terms:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
3. Now, combine like terms:
- Combine the [tex]\(x^4\)[/tex] terms:
[tex]\[
3x^4
\][/tex]
- Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
-9x^3 - 4x^3 = -13x^3
\][/tex]
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
6x^2 + 12x^2 + 5x^2 = 23x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-8x - 15x = -23x
\][/tex]
- The constant term:
[tex]\[
10
\][/tex]
So, after combining all the like terms, the final result is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
The correct answer is option B: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
1. Distribute each term from the first polynomial across each term from the second polynomial:
- First term of [tex]\(3x^2\)[/tex] with all terms of the second polynomial:
[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 of [tex]\(-4x\)[/tex] with all terms of the second polynomial:
[tex]\[
-4x \cdot x^2 = -4x^3
\][/tex]
[tex]\[
-4x \cdot (-3x) = 12x^2
\][/tex]
[tex]\[
-4x \cdot 2 = -8x
\][/tex]
- Third term of [tex]\(5\)[/tex] with all terms of the second polynomial:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-3x) = -15x
\][/tex]
[tex]\[
5 \cdot 2 = 10
\][/tex]
2. Combine all the distributed terms:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
3. Now, combine like terms:
- Combine the [tex]\(x^4\)[/tex] terms:
[tex]\[
3x^4
\][/tex]
- Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
-9x^3 - 4x^3 = -13x^3
\][/tex]
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
6x^2 + 12x^2 + 5x^2 = 23x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-8x - 15x = -23x
\][/tex]
- The constant term:
[tex]\[
10
\][/tex]
So, after combining all the like terms, the final result is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
The correct answer is option B: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
