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Magazine Chapter 9: Problem 22
Decompose into partial fractions. Check your answers using a graphing calculator. $$\frac{26 x^{2}-36 x+22}{(x-4)(2 x-1)^{2}}$$
Short Answer
Expert verified
The partial fraction decomposition is \( \frac{5}{x-4} + \frac{3}{2x-1} + \frac{1}{(2x-1)^2} \).
Step by step solution
01
Identify the Form of the Partial Fractions
Recognize that the denominator \( (x-4)(2x-1)^{2} \) indicates the form of partial fractions. Write the generic form as \[ \frac{A}{x-4} + \frac{B}{2x-1} + \frac{C}{(2x-1)^2} \]
02
Rewrite the Equation
Express the given fraction in terms of the partial fractions: \[ \frac{26 x^{2}-36 x+22}{(x-4)(2 x-1)^{2}} = \frac{A}{x-4} + \frac{B}{2x-1} + \frac{C}{(2x-1)^2} \]
03
Clear the Denominator
Multiply both sides by the denominator \( (x-4)(2x-1)^2 \) to get: \[ 26 x^2 - 36 x + 22 = A(2x-1)^2 + B(x-4)(2x-1) + C(x-4) \]
04
Expand and Combine Like Terms
Expand \( (2x-1)^2 \) as \[ 4x^2 - 4x + 1 \] and distribute \( A \): \[ A(4x^2 - 4x + 1) + B(x-4)(2x-1) + C(x-4) \]. Then expand and collect terms.
05
Set Up a System of Equations
Match the coefficients of corresponding powers of \( x \) on both sides: \[ 26x^2 - 36x + 22 = 4Ax^2 - 4Ax + A + 2Bx^2 - 9Bx + 4B + Cx - 4C \]. Group terms to give: \[ 26x^2 - 36x + 22 = (4A + 2B)x^2 + (-4A - 9B + C)x + (A + 4B - 4C) \]
06
Solve the System of Equations
From the coefficients, set up the system of equations: \[ 4A + 2B = 26 \], \[ -4A - 9B + C = -36 \], and \[ A + 4B - 4C = 22 \]. Solve these equations to find \( A, B, \) and \( C \): \( A = 5 \), \( B = 3 \), \( C = 1 \)
07
Write the Partial Fraction Decomposition
Substitute \( A, B, \) and \( C \) back into the partial fractions: \[ \frac{5}{x-4} + \frac{3}{2x-1} + \frac{1}{(2x-1)^2} \]
08
Check Using a Graphing Calculator
Verify the decomposition by graphing both sides of: \[ \frac{26 x^{2}-36 x+22}{(x-4)(2 x-1)^{2}} \] and \[ \frac{5}{x-4} + \frac{3}{2x-1} + \frac{1}{(2x-1)^2} \]. Ensure both graphs are identical.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Decomposing Rational Functions
Decomposing rational functions involves breaking down a complex fraction into simpler fractions that are easier to manage. The initial step is to recognize the form of the partial fractions. For example, given the fraction \( \frac{26 x^{2}-36 x+22}{(x-4)(2 x-1)^{2}} \), identify that the denominator consists of \( (x-4) \) and \( (2x-1)^2 \). This tells us that our partial fractions will be of the form: \( \frac{A}{x-4} + \frac{B}{2x-1} + \frac{C}{(2x-1)^2} \).
Next, rewrite the initial fraction in terms of these partial fractions and multiply both sides by the denominator \( (x-4)(2x-1)^2 \) to clear it. This results in a polynomial equation that can be solved for the coefficients \( A, B, \) and \( C \).
Next, rewrite the initial fraction in terms of these partial fractions and multiply both sides by the denominator \( (x-4)(2x-1)^2 \) to clear it. This results in a polynomial equation that can be solved for the coefficients \( A, B, \) and \( C \).
System of Equations
To find the values of coefficients \( A, B, \) and \( C \), set up a system of equations by matching the coefficients of like terms from both sides of the polynomial equation.
For instance, from our expanded equation \( 26 x^{2}-36 x+22 = 4Ax^{2} -4Ax + A + 2Bx^{2}-9Bx + 4B + Cx - 4C \), group and match the coefficients of \( x^{2} \), \( x \), and the constant.
This gives us the system:
For instance, from our expanded equation \( 26 x^{2}-36 x+22 = 4Ax^{2} -4Ax + A + 2Bx^{2}-9Bx + 4B + Cx - 4C \), group and match the coefficients of \( x^{2} \), \( x \), and the constant.
This gives us the system:
- \( 4A + 2B = 26 \)
- \( -4A - 9B + C = -36 \)
- \( A + 4B - 4C = 22 \)
Graphing Calculator Verification
Using a graphing calculator helps to verify the correctness of the partial fraction decomposition.
Graph the original function \( \frac{26 x^{2}-36 x+22}{(x-4)(2 x-1)^{2}} \) and the decomposed form \( \frac{5}{x-4} + \frac{3}{2x-1} + \frac{1}{(2x-1)^2} \). Both graphs should be identical if the decomposition is correct.
This visual confirmation is crucial as it reinforces the algebraic solution with a graphical representation, ensuring accuracy.
Graph the original function \( \frac{26 x^{2}-36 x+22}{(x-4)(2 x-1)^{2}} \) and the decomposed form \( \frac{5}{x-4} + \frac{3}{2x-1} + \frac{1}{(2x-1)^2} \). Both graphs should be identical if the decomposition is correct.
This visual confirmation is crucial as it reinforces the algebraic solution with a graphical representation, ensuring accuracy.
Polynomial Expansion
Polynomial expansion simplifies expressions and is vital in partial fractions. In our example, expand \( (2x-1)^2 \) to get \( 4x^{2} - 4x + 1 \). Then, multiply this with the coefficient \( A \) to distribute through the terms.
Additionally, address each partial fraction term by distributing and combining like terms. This step-by-step expansion and combination make it easier to set up the system of equations required to solve for \( A, B, \) and \( C \).
Understanding each polynomial term's role and how they combine builds a strong foundation for solving rational functions through partial fractions.
Additionally, address each partial fraction term by distributing and combining like terms. This step-by-step expansion and combination make it easier to set up the system of equations required to solve for \( A, B, \) and \( C \).
Understanding each polynomial term's role and how they combine builds a strong foundation for solving rational functions through partial fractions.
