Find the partial fraction decomposition of x⁴+x³+x²+1/x²+x-2

To find the partial fraction decomposition of the rational function:

\[
\frac{x^4 + x^3 + x^2 + 1}{x^2 + x - 2}
\]

### Step 1: Factor the denominator
We start by factoring the denominator \( x^2 + x - 2 \). To do this, find the factors of \(-2\) that sum up to \(1\) (the coefficient of \(x\)):

\[
x^2 + x - 2 = (x - 1)(x + 2)
\]

### Step 2: Express the function as a sum of partial fractions
Now we express the given rational function as a sum of partial fractions with the factored denominator:

\[
\frac{x^4 + x^3 + x^2 + 1}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}
\]

### Step 3: Clear the denominator
Multiply both sides of the equation by the denominator \((x - 1)(x + 2)\) to get rid of the denominator:

\[
x^4 + x^3 + x^2 + 1 = A(x + 2) + B(x - 1)
\]

### Step 4: Expand and simplify
Now expand the right-hand side:

\[
x^4 + x^3 + x^2 + 1 = A(x + 2) + B(x - 1)
\]
\[
= A(x) + A(2) + B(x) - B
\]
\[
= (A + B)x + (2A - B)
\]

### Step 5: Compare coefficients
Now, compare the coefficients of powers of \(x\) on both sides of the equation. For this expression, we match the coefficients of like powers of \(x\) on both sides.



This page provides a detailed step-by-step solution for finding the partial fraction decomposition of the rational function x⁴ + x³ + x² + 1 over the factored denominator x² + x - 2. The solution includes factorization techniques and comparing coefficients, ideal for students in Grades 10-12.

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