Answer :
Sure! Let's multiply the two polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex]. We'll use distributive property (also known as FOIL when applied to binomials) to find the product step-by-step.
Given:
[tex]\[
(3x^2 - 4x + 5)(x^2 - 3x + 2)
\][/tex]
### Step 1: Multiply each term in the first polynomial by each term in the second polynomial
First, multiply [tex]\(3x^2\)[/tex] by the terms in 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]
Next, multiply [tex]\(-4x\)[/tex] by the terms in the second polynomial:
[tex]\[
-4x \cdot x^2 = -4x^3
\][/tex]
[tex]\[
-4x \cdot (-3x) = 12x^2
\][/tex]
[tex]\[
-4x \cdot 2 = -8x
\][/tex]
Then, multiply [tex]\(5\)[/tex] by the terms in the second polynomial:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-3x) = -15x
\][/tex]
[tex]\[
5 \cdot 2 = 10
\][/tex]
### Step 2: Combine all the terms
Now, let's combine all the terms we've obtained:
[tex]\[
3x^4 + (-9x^3) + 6x^2 + (-4x^3) + 12x^2 + (-8x) + 5x^2 + (-15x) + 10
\][/tex]
### Step 3: Combine like terms
Combine like terms:
[tex]\[
3x^4
\][/tex]
[tex]\[
-9x^3 - 4x^3 = -13x^3
\][/tex]
[tex]\[
6x^2 + 12x^2 + 5x^2 = 23x^2
\][/tex]
[tex]\[
-8x - 15x = -23x
\][/tex]
[tex]\[
10
\][/tex]
### Step 4: Write the final polynomial
Combine all these results to get the final polynomial:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
This matches option D. So the correct answer is:
[tex]\[
\boxed{D. \; 3x^4 - 13x^3 + 23x^2 - 23x + 10}
\][/tex]
Given:
[tex]\[
(3x^2 - 4x + 5)(x^2 - 3x + 2)
\][/tex]
### Step 1: Multiply each term in the first polynomial by each term in the second polynomial
First, multiply [tex]\(3x^2\)[/tex] by the terms in 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]
Next, multiply [tex]\(-4x\)[/tex] by the terms in the second polynomial:
[tex]\[
-4x \cdot x^2 = -4x^3
\][/tex]
[tex]\[
-4x \cdot (-3x) = 12x^2
\][/tex]
[tex]\[
-4x \cdot 2 = -8x
\][/tex]
Then, multiply [tex]\(5\)[/tex] by the terms in the second polynomial:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-3x) = -15x
\][/tex]
[tex]\[
5 \cdot 2 = 10
\][/tex]
### Step 2: Combine all the terms
Now, let's combine all the terms we've obtained:
[tex]\[
3x^4 + (-9x^3) + 6x^2 + (-4x^3) + 12x^2 + (-8x) + 5x^2 + (-15x) + 10
\][/tex]
### Step 3: Combine like terms
Combine like terms:
[tex]\[
3x^4
\][/tex]
[tex]\[
-9x^3 - 4x^3 = -13x^3
\][/tex]
[tex]\[
6x^2 + 12x^2 + 5x^2 = 23x^2
\][/tex]
[tex]\[
-8x - 15x = -23x
\][/tex]
[tex]\[
10
\][/tex]
### Step 4: Write the final polynomial
Combine all these results to get the final polynomial:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
This matches option D. So the correct answer is:
[tex]\[
\boxed{D. \; 3x^4 - 13x^3 + 23x^2 - 23x + 10}
\][/tex]
