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Sagot :
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.
### 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.
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