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Answer :
Certainly! Let's work through the problem step by step.
We want to multiply:
[tex]\[ (3x^2 - 4x + 5)(x^2 - 3x + 2) \][/tex]
To multiply these polynomials, we will use the distributive property (sometimes called the FOIL method for binomials). This involves distributing each term in the first polynomial to every term in the second polynomial.
Steps:
1. Distribute [tex]\(3x^2\)[/tex] through each term in [tex]\((x^2 - 3x + 2)\)[/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]
2. Distribute [tex]\(-4x\)[/tex] through each term in [tex]\((x^2 - 3x + 2)\)[/tex]:
[tex]\[
-4x \cdot x^2 = -4x^3
\][/tex]
[tex]\[
-4x \cdot (-3x) = 12x^2
\][/tex]
[tex]\[
-4x \cdot 2 = -8x
\][/tex]
3. Distribute [tex]\(5\)[/tex] through each term in [tex]\((x^2 - 3x + 2)\)[/tex]:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-3x) = -15x
\][/tex]
[tex]\[
5 \cdot 2 = 10
\][/tex]
Now, we combine all these partial results:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
Next, we combine like terms (terms with the same power of [tex]\(x\)[/tex]):
[tex]\[
3x^4 + (-9x^3 - 4x^3) + (6x^2 + 12x^2 + 5x^2) + (-8x - 15x) + 10
\][/tex]
Simplify these:
[tex]\[
3x^4 + (-13x^3) + (23x^2) + (-23x) + 10
\][/tex]
So, the result of the multiplication is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C. \, 3x^4 - 13x^3 + 23x^2 - 23x + 10}
\][/tex]
We want to multiply:
[tex]\[ (3x^2 - 4x + 5)(x^2 - 3x + 2) \][/tex]
To multiply these polynomials, we will use the distributive property (sometimes called the FOIL method for binomials). This involves distributing each term in the first polynomial to every term in the second polynomial.
Steps:
1. Distribute [tex]\(3x^2\)[/tex] through each term in [tex]\((x^2 - 3x + 2)\)[/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]
2. Distribute [tex]\(-4x\)[/tex] through each term in [tex]\((x^2 - 3x + 2)\)[/tex]:
[tex]\[
-4x \cdot x^2 = -4x^3
\][/tex]
[tex]\[
-4x \cdot (-3x) = 12x^2
\][/tex]
[tex]\[
-4x \cdot 2 = -8x
\][/tex]
3. Distribute [tex]\(5\)[/tex] through each term in [tex]\((x^2 - 3x + 2)\)[/tex]:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-3x) = -15x
\][/tex]
[tex]\[
5 \cdot 2 = 10
\][/tex]
Now, we combine all these partial results:
[tex]\[
3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10
\][/tex]
Next, we combine like terms (terms with the same power of [tex]\(x\)[/tex]):
[tex]\[
3x^4 + (-9x^3 - 4x^3) + (6x^2 + 12x^2 + 5x^2) + (-8x - 15x) + 10
\][/tex]
Simplify these:
[tex]\[
3x^4 + (-13x^3) + (23x^2) + (-23x) + 10
\][/tex]
So, the result of the multiplication is:
[tex]\[
3x^4 - 13x^3 + 23x^2 - 23x + 10
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C. \, 3x^4 - 13x^3 + 23x^2 - 23x + 10}
\][/tex]
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