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Multiply the following using the vertical multiplication method:

[tex]\[
\begin{array}{r}
x^2 + 5x + 1 \\
\times (3x^2 - 2x + 4) \\
\hline
\end{array}
\][/tex]

A. [tex]\(3x^4 - 13x^3 + 3x^2 + 18x + 4\)[/tex]
B. [tex]\(3x^4 + x^3 + x^2 + 18x + 4\)[/tex]
C. [tex]\(3x^4 - x^3 + 3x^2 + x + 4\)[/tex]
D. [tex]\(3x^4 + 13x^3 - 3x^2 + 18x + 4\)[/tex]


Sagot :

Sure, let's start by performing the polynomial multiplication [tex]\( (x^2 + 5x + 1) \times (3x^2 - 2x + 4) \)[/tex] using the vertical multiplication method.

1. First, distribute the term [tex]\( x^2 \)[/tex] over each term in [tex]\( 3x^2 - 2x + 4 \)[/tex]:
[tex]\[ x^2 \cdot 3x^2 = 3x^4 \][/tex]
[tex]\[ x^2 \cdot (-2x) = -2x^3 \][/tex]
[tex]\[ x^2 \cdot 4 = 4x^2 \][/tex]

So the result of [tex]\( x^2 \cdot (3x^2 - 2x + 4) \)[/tex] is:
[tex]\[ 3x^4 - 2x^3 + 4x^2 \][/tex]

2. Next, distribute the term [tex]\( 5x \)[/tex] over each term in [tex]\( 3x^2 - 2x + 4 \)[/tex]:
[tex]\[ 5x \cdot 3x^2 = 15x^3 \][/tex]
[tex]\[ 5x \cdot (-2x) = -10x^2 \][/tex]
[tex]\[ 5x \cdot 4 = 20x \][/tex]

So the result of [tex]\( 5x \cdot (3x^2 - 2x + 4) \)[/tex] is:
[tex]\[ 15x^3 - 10x^2 + 20x \][/tex]

3. Finally, distribute the term [tex]\( 1 \)[/tex] over each term in [tex]\( 3x^2 - 2x + 4 \)[/tex]:
[tex]\[ 1 \cdot 3x^2 = 3x^2 \][/tex]
[tex]\[ 1 \cdot (-2x) = -2x \][/tex]
[tex]\[ 1 \cdot 4 = 4 \][/tex]

So the result of [tex]\( 1 \cdot (3x^2 - 2x + 4) \)[/tex] is:
[tex]\[ 3x^2 - 2x + 4 \][/tex]

Now, let's add all these results together, aligning them by their degree of [tex]\( x \)[/tex]:

[tex]\[ \begin{aligned} &\ 3x^4 \\ &- 2x^3 + 15x^3 = 13x^3 \\ &4x^2 - 10x^2 + 3x^2 = -3x^2 \\ &20x - 2x = 18x \\ &\ 4 \end{aligned} \][/tex]

So the final result of multiplying [tex]\( x^2 + 5x + 1 \)[/tex] and [tex]\( 3x^2 - 2x + 4 \)[/tex] is:
[tex]\[ 3x^4 + 13x^3 - 3x^2 + 18x + 4 \][/tex]

Therefore, the correct answer is:
D. [tex]\( 3x^4 + 13 x^3 - 3 x^2 + 18 x + 4 \)[/tex]