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Sagot :
Certainly! Let's go through the problem step-by-step to rewrite the improper rational expression as the sum of a polynomial and a proper rational expression, and then find the partial fraction decomposition of the proper rational expression.
Given:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} \][/tex]
### Step 1: Perform Polynomial Division
First, we'll perform polynomial division to divide [tex]\(x^4 - 6x^2 + x - 16\)[/tex] by [tex]\(x^2 + 8x + 16\)[/tex]. This process can be summarized as follows:
1. Divide the leading term of the numerator [tex]\(x^4\)[/tex] by the leading term of the denominator [tex]\(x^2\)[/tex] to get [tex]\(x^2\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by the entire denominator [tex]\((x^2 + 8x + 16)\)[/tex] to get [tex]\(x^4 + 8x^3 + 16x^2\)[/tex].
3. Subtract this result from the original numerator to get the new polynomial:
[tex]\[ (x^4 - 6x^2 + x - 16) - (x^4 + 8x^3 + 16x^2) = -8x^3 - 22x^2 + x - 16. \][/tex]
4. Repeat this process with [tex]\(-8x^3\)[/tex] divided by [tex]\(x^2\)[/tex], giving [tex]\(-8x\)[/tex]. We multiply [tex]\(-8x\)[/tex] by the entire denominator to get [tex]\(-8x^3 - 64x^2 - 128x\)[/tex].
5. Subtract this to get the next polynomial and continue until we obtain a remainder that has a degree smaller than the degree of the denominator.
Here's the summarized result of the polynomial division:
The quotient is:
[tex]\[ x^2 - 8x + 42 \][/tex]
The remainder is:
[tex]\[ -207x - 688 \][/tex]
Thus, we can rewrite the expression as:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} = x^2 - 8x + 42 + \frac{-207x - 688}{x^2 + 8x + 16} \][/tex]
### Step 2: Partial Fraction Decomposition of the Proper Rational Expression
Now we decompose [tex]\(\frac{-207x - 688}{x^2 + 8x + 16}\)[/tex] into partial fractions. Note that [tex]\(x^2 + 8x + 16\)[/tex] can be factored as [tex]\((x + 4)^2\)[/tex].
Thus the decomposition looks like:
[tex]\[ \frac{-207x - 688}{(x + 4)^2} = \frac{A}{x + 4} + \frac{B}{(x + 4)^2} \][/tex]
Equate the numerators:
[tex]\[ -207x - 688 = A(x + 4) + B \][/tex]
Solving for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ -207x - 688 = Ax + 4A + B \][/tex]
We equate the coefficients of like terms:
1. For [tex]\(x\)[/tex]:
[tex]\[ -207 = A \][/tex]
2. For the constant term:
[tex]\[ -688 = 4A + B \][/tex]
Substituting [tex]\(A = -207\)[/tex] into [tex]\(4A + B\)[/tex]:
[tex]\[ -688 = 4(-207) + B \implies -688 = -828 + B \implies B = 140 \][/tex]
So, the partial fraction decomposition is:
[tex]\[ \frac{-207x - 688}{(x + 4)^2} = \frac{-207}{x + 4} + \frac{140}{(x + 4)^2} \][/tex]
### Step 3: Express the Improper Rational Expression
Combining everything, we can rewrite the original improper rational expression as:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} = x^2 - 8x + 42 + \left( \frac{-207}{x + 4} + \frac{140}{(x + 4)^2} \right) \][/tex]
So, the final expression is:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} = x^2 - 8x + 42 + \frac{-207}{x + 4} + \frac{140}{(x + 4)^2} \][/tex]
This completes the step-by-step solution!
Given:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} \][/tex]
### Step 1: Perform Polynomial Division
First, we'll perform polynomial division to divide [tex]\(x^4 - 6x^2 + x - 16\)[/tex] by [tex]\(x^2 + 8x + 16\)[/tex]. This process can be summarized as follows:
1. Divide the leading term of the numerator [tex]\(x^4\)[/tex] by the leading term of the denominator [tex]\(x^2\)[/tex] to get [tex]\(x^2\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by the entire denominator [tex]\((x^2 + 8x + 16)\)[/tex] to get [tex]\(x^4 + 8x^3 + 16x^2\)[/tex].
3. Subtract this result from the original numerator to get the new polynomial:
[tex]\[ (x^4 - 6x^2 + x - 16) - (x^4 + 8x^3 + 16x^2) = -8x^3 - 22x^2 + x - 16. \][/tex]
4. Repeat this process with [tex]\(-8x^3\)[/tex] divided by [tex]\(x^2\)[/tex], giving [tex]\(-8x\)[/tex]. We multiply [tex]\(-8x\)[/tex] by the entire denominator to get [tex]\(-8x^3 - 64x^2 - 128x\)[/tex].
5. Subtract this to get the next polynomial and continue until we obtain a remainder that has a degree smaller than the degree of the denominator.
Here's the summarized result of the polynomial division:
The quotient is:
[tex]\[ x^2 - 8x + 42 \][/tex]
The remainder is:
[tex]\[ -207x - 688 \][/tex]
Thus, we can rewrite the expression as:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} = x^2 - 8x + 42 + \frac{-207x - 688}{x^2 + 8x + 16} \][/tex]
### Step 2: Partial Fraction Decomposition of the Proper Rational Expression
Now we decompose [tex]\(\frac{-207x - 688}{x^2 + 8x + 16}\)[/tex] into partial fractions. Note that [tex]\(x^2 + 8x + 16\)[/tex] can be factored as [tex]\((x + 4)^2\)[/tex].
Thus the decomposition looks like:
[tex]\[ \frac{-207x - 688}{(x + 4)^2} = \frac{A}{x + 4} + \frac{B}{(x + 4)^2} \][/tex]
Equate the numerators:
[tex]\[ -207x - 688 = A(x + 4) + B \][/tex]
Solving for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ -207x - 688 = Ax + 4A + B \][/tex]
We equate the coefficients of like terms:
1. For [tex]\(x\)[/tex]:
[tex]\[ -207 = A \][/tex]
2. For the constant term:
[tex]\[ -688 = 4A + B \][/tex]
Substituting [tex]\(A = -207\)[/tex] into [tex]\(4A + B\)[/tex]:
[tex]\[ -688 = 4(-207) + B \implies -688 = -828 + B \implies B = 140 \][/tex]
So, the partial fraction decomposition is:
[tex]\[ \frac{-207x - 688}{(x + 4)^2} = \frac{-207}{x + 4} + \frac{140}{(x + 4)^2} \][/tex]
### Step 3: Express the Improper Rational Expression
Combining everything, we can rewrite the original improper rational expression as:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} = x^2 - 8x + 42 + \left( \frac{-207}{x + 4} + \frac{140}{(x + 4)^2} \right) \][/tex]
So, the final expression is:
[tex]\[ \frac{x^4 - 6x^2 + x - 16}{x^2 + 8x + 16} = x^2 - 8x + 42 + \frac{-207}{x + 4} + \frac{140}{(x + 4)^2} \][/tex]
This completes the step-by-step solution!
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