Answered

Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

Multiply the following using the vertical multiplication method:

[tex]\[
\begin{array}{l}
x^2-5x+1 \\
x^2+2x+4
\end{array}
\][/tex]

A. [tex]\(3x^4 + x^3 + 10x^2 - 18x + 4\)[/tex]

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

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

D. [tex]\(3x^4 + x^3 + 10x^2 + x + 4\)[/tex]

Sagot :

GinnyAnswer
To multiply the polynomials [tex]\(P(x) = x^2 - 5x + 1\)[/tex] and [tex]\(Q(x) = x^2 + 2x + 4\)[/tex], we will use the vertical multiplication method. Let's break it down step-by-step:

### Step 1: Multiply each term in [tex]\(P(x)\)[/tex] by each term in [tex]\(Q(x)\)[/tex]

First, distribute [tex]\(x^2\)[/tex] from [tex]\(Q(x)\)[/tex]:
[tex]\[ (x^2 \cdot (x^2 - 5x + 1)) = x^4 - 5x^3 + x^2 \][/tex]

Next, distribute [tex]\(2x\)[/tex] from [tex]\(Q(x)\)[/tex]:
[tex]\[ (2x \cdot (x^2 - 5x + 1)) = 2x^3 - 10x^2 + 2x \][/tex]

Finally, distribute [tex]\(4\)[/tex] from [tex]\(Q(x)\)[/tex]:
[tex]\[ (4 \cdot (x^2 - 5x + 1)) = 4x^2 - 20x + 4 \][/tex]

### Step 2: Add the results together, combining like terms

To combine the results, we align the terms with the same degree:

[tex]\[ \begin{array}{rcccccc} & x^4 & - 5x^3 & + x^2 & & & \\ + & & 2x^3 & - 10x^2 & + 2x & & \\ + & & & 4x^2 & - 20x & + 4 & \\ \hline \end{array} \][/tex]

Now, let's sum these expressions:

1. [tex]\(x^4\)[/tex] terms:
[tex]\[ x^4 \][/tex]

2. [tex]\(x^3\)[/tex] terms:
[tex]\[ -5x^3 + 2x^3 = -3x^3 \][/tex]

3. [tex]\(x^2\)[/tex] terms:
[tex]\[ x^2 - 10x^2 + 4x^2 = -5x^2 \][/tex]

4. [tex]\(x\)[/tex] terms:
[tex]\[ 2x - 20x = -18x \][/tex]

5. Constant terms:
[tex]\[ +4 \][/tex]

So, when we combine all like terms, we get:

[tex]\[ x^4 - 3x^3 - 5x^2 - 18x + 4 \][/tex]

### Step 3: Compare the result with the given choices

Let's rewrite the result:
[tex]\[ x^4 - 3x^3 - 5x^2 - 18x + 4 \][/tex]

Now, let’s check which provided choice matches our result:

A. [tex]\(3 x^4 + x^3 + 10 x^2 - 18 x + 4\)[/tex]

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

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

D. [tex]\(3 x^4 + x^3 + 10 x^2 + x + 4\)[/tex]

None of the provided options exactly match our result. Therefore, there might be an error. Let's re-evaluate our combination of like terms to ensure accuracy.

Correctly combining, our result should be:
[tex]\[ x^4 - 3x^3 - 5x^2 - 18x + 4 \][/tex]

Upon reviewing my work, it seems original calculations were correct. Thus, the correct answer should ideally match [tex]\( x^4 - 3x^3 - 5x^2 - 18x + 4 \)[/tex]. Based on given options, recheck provided choices; logically no option suits exactly our derived value properly.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.