QuestionJuly 15, 2025

The partial fraction decomposition of (x^2+65)/(x^3)+x^(2) can be written in the form of (f(x))/(x)+(g(x))/(x^2)+(h(x))/(x+1) where f(x)=square g(x)=square h(x)=square

The partial fraction decomposition of (x^2+65)/(x^3)+x^(2) can be written in the form of (f(x))/(x)+(g(x))/(x^2)+(h(x))/(x+1) where f(x)=square g(x)=square h(x)=square
The partial fraction decomposition of (x^2+65)/(x^3)+x^(2) can be written in the form of (f(x))/(x)+(g(x))/(x^2)+(h(x))/(x+1) where f(x)=square g(x)=square h(x)=square

Solution
4.7(255 votes)

Answer

f(x) = -65, \quad g(x) = 65, \quad h(x) = 66 Explanation 1. Factor the Denominator The denominator x^3 + x^2 factors as x^2(x+1). 2. Set Up Partial Fractions Express \frac{x^2 + 65}{x^3 + x^2} as \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}. 3. Clear the Denominator Multiply through by x^2(x+1) to get x^2 + 65 = A x (x+1) + B (x+1) + C x^2. 4. Expand and Collect Terms Expand: Ax^2 + Ax + Bx + B + Cx^2. Combine: (A+C)x^2 + (A+B)x + B. 5. Equate Coefficients Compare with x^2 + 65: - A + C = 1 - A + B = 0 - B = 65 6. Solve for Constants From B = 65, A + 65 = 0 \Rightarrow A = -65. Then A + C = 1 \Rightarrow -65 + C = 1 \Rightarrow C = 66.

Explanation

1. Factor the Denominator<br /> The denominator $x^3 + x^2$ factors as $x^2(x+1)$.<br /><br />2. Set Up Partial Fractions<br /> Express $\frac{x^2 + 65}{x^3 + x^2}$ as $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1}$.<br /><br />3. Clear the Denominator<br /> Multiply through by $x^2(x+1)$ to get $x^2 + 65 = A x (x+1) + B (x+1) + C x^2$.<br /><br />4. Expand and Collect Terms<br /> Expand: $Ax^2 + Ax + Bx + B + Cx^2$. Combine: $(A+C)x^2 + (A+B)x + B$.<br /><br />5. Equate Coefficients<br /> Compare with $x^2 + 65$: <br />- $A + C = 1$<br />- $A + B = 0$<br />- $B = 65$<br /><br />6. Solve for Constants<br /> From $B = 65$, $A + 65 = 0 \Rightarrow A = -65$. Then $A + C = 1 \Rightarrow -65 + C = 1 \Rightarrow C = 66$.
Click to rate:

Similar Questions