Math Problem Statement

Determine the constants A, B, and C in the partial fraction decomposition: \[ \frac{4x^2 + 3}{(x - 1)(x^2 + x + 5)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + x + 5}. \]

Solution

To solve this problem, we need to determine the constants AA, BB, and CC in the equation:

4x2+3(x−1)(x2+x+5)=Ax−1+Bx+Cx2+x+5.\frac{4x^2 + 3}{(x - 1)(x^2 + x + 5)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 + x + 5}.

Step 1: Rewrite the Equation

  1. The given equation already provides the structure of a partial fraction decomposition.
  2. For the equation to be true, the left and right sides must be equal for all values of xx, which means their numerators must match once they are combined over a common denominator.

Step 2: Finding the Least Common Denominator (LCD)

To combine the fractions on the right-hand side, we use the common denominator:

LCD=(x−1)(x2+x+5).\text{LCD} = (x - 1)(x^2 + x + 5).

Step 3: Set Up the Equation

By rewriting both sides with this common denominator, we can equate the numerators:

4x2+3=A(x2+x+5)+(Bx+C)(x−1).4x^2 + 3 = A(x^2 + x + 5) + (Bx + C)(x - 1).

Step 4: Expand and Simplify the Right-Hand Side

Now, we expand each term on the right-hand side:

  1. Expand A(x2+x+5)A(x^2 + x + 5): A(x2+x+5)=Ax2+Ax+5A.A(x^2 + x + 5) = Ax^2 + Ax + 5A.

  2. Expand (Bx+C)(x−1)(Bx + C)(x - 1): (Bx+C)(x−1)=Bx2−Bx+Cx−C=Bx2+(C−B)x−C.(Bx + C)(x - 1) = Bx^2 - Bx + Cx - C = Bx^2 + (C - B)x - C.

Now, combining all terms on the right side, we get:

4x2+3=(A+B)x2+(A+C−B)x+(5A−C).4x^2 + 3 = (A + B)x^2 + (A + C - B)x + (5A - C).

Step 5: Equate Coefficients

Now we equate the coefficients of x2x^2, xx, and the constant term on both sides:

  1. For x2x^2: A+B=4A + B = 4.
  2. For xx: A+C−B=0A + C - B = 0.
  3. For the constant: 5A−C=35A - C = 3.

This forms a system of equations:

A + B = 4, \\ A + C - B = 0, \\ 5A - C = 3. \end{cases}$$ ### Next Steps 1. Solve this system of equations to find $$A$$, $$B$$, and $$C$$. 2. Once we have $$A$$, $$B$$, and $$C$$, we can substitute them back into the partial fraction decomposition. Would you like to continue with solving this system of equations, or do you need further clarification on any part of the process so far? --- Here are five related questions that might deepen your understanding: 1. How do we find the partial fraction decomposition for a different denominator, such as $$(x + 2)(x^2 + 3x + 2)$$? 2. What is the general form of partial fraction decomposition for a fraction with higher-order polynomials in the denominator? 3. Why do we only consider terms of $$x^2$$, $$x$$, and the constant for matching coefficients? 4. How would the process change if there were complex roots in the denominator? 5. What other methods, besides equating coefficients, could we use to solve for $$A$$, $$B$$, and $$C$$? **Tip:** When solving for unknowns in partial fraction decomposition, equating coefficients after expanding terms is an efficient way to match terms accurately.

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Math Problem Analysis

Mathematical Concepts

Algebra
Partial Fraction Decomposition
Polynomial Division

Formulas

Partial fraction decomposition formula
System of equations for matching coefficients

Theorems

Fundamental Theorem of Algebra (for decomposition)

Suitable Grade Level

Grades 10-12

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