Split Using Partial Fraction Decomposition (8y+7)/(y^2+y-2)
The given problem is asking you to perform partial fraction decomposition on the algebraic expression (8y+7)/(y^2+y-2). Partial fraction decomposition is a technique used in algebra and calculus to break down a complex rational expression (a fraction where the numerator and the denominator are polynomials) into several simpler fractions that are easier to work with. This process is especially helpful in integrating rational functions. The expression given has a quadratic polynomial in the denominator, and the aim is to express it as a sum of fractions with linear denominators.
$\frac{8 y + 7}{y^{2} + y - 2}$
Answer
Solution:
Step:1
Begin by breaking down the given fraction and multiplying by the common denominator.
Step:1.1
Factor $y^{2} + y - 2$ by searching for two numbers that multiply to give $c$ and add to give $b$. Here, we need numbers that multiply to $-2$ and add to $1$. The numbers are $-1$ and $2$.
Step:1.1.1
The factored form is $\frac{8y + 7}{(y - 1)(y + 2)}$.
Step:1.2
Assign a variable for the numerator of the new fraction for each denominator factor, such as $A$ for $\frac{A}{y - 1}$.
Step:1.3
Similarly, assign a variable $B$ for the other fraction, resulting in $\frac{A}{y - 1} + \frac{B}{y + 2}$.
Step:1.4
Multiply the entire equation by the original denominator $(y - 1)(y + 2)$ to clear the denominators.
Step:1.5
Eliminate the common factor $(y - 1)$.
Step:1.5.1
The equation simplifies to $\frac{(8y + 7)(y + 2)}{y + 2}$.
Step:1.6
Now, remove the common factor $(y + 2)$.
Step:1.6.1
We are left with $8y + 7 = \frac{A(y + 2)}{y - 1} + \frac{B(y - 1)}{y + 2}$.
Step:1.7
Simplify each term by distributing and combining like terms.
Step:1.7.1
The equation simplifies to $8y + 7 = Ay + 2A + By - B$.
Step:1.8
Combine like terms to get $8y + 7 = (A + B)y + (2A - B)$.
Step:2
Formulate a system of equations based on the coefficients of $y$ and the constant terms.
Step:2.1
For the coefficients of $y$, we have $8 = A + B$.
Step:2.2
For the constant terms, we have $7 = 2A - B$.
Step:3
Solve the system of equations to find the values of $A$ and $B$.
Step:3.1
Isolate $A$ in the equation $8 = A + B$ to get $A = 8 - B$.
Step:3.2
Substitute $A$ with $8 - B$ in the second equation to find $B$.
Step:3.3
Solve for $B$ to get $B = 3$.
Step:3.4
Substitute $B$ back into the equation for $A$ to find $A = 5$.
Step:4
Insert the values of $A$ and $B$ back into the partial fractions to get $\frac{5}{y - 1} + \frac{3}{y + 2}$.
Knowledge Notes:
Partial fraction decomposition is a technique used to express a rational function as the sum of simpler fractions. It is particularly useful when integrating rational functions. The process involves the following steps:
Factor the denominator of the rational function.
Write the partial fraction decomposition using unknown coefficients for the numerators of each fraction corresponding to the factors of the denominator.
Multiply through by the common denominator to clear the fractions.
Simplify the resulting equation and match coefficients on both sides to form a system of linear equations.
Solve the system of equations to find the values of the unknown coefficients.
Substitute the coefficients back into the partial fractions to complete the decomposition.
In the given problem, we applied these steps to decompose the fraction $\frac{8y + 7}{y^2 + y - 2}$ into partial fractions $\frac{5}{y - 1} + \frac{3}{y + 2}$.
