Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$ \frac{3 x^{2}-7 x+33}{x^{3}+27} $$

Step-by-Step Solution

Answer

Decompose: \(\frac{3}{x+3} + \frac{2}{x^2 - 3x + 9}\).

Step 1: Factorize the Denominator

The given denominator is a sum of cubes since it can be expressed as \( x^3 + 27 = x^3 + 3^3 \). The formula for the sum of cubes is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Hence, \( x^3 + 27 \) can be factorized as \( (x + 3)(x^2 - 3x + 9) \).

Step 2: Set up the Partial Fraction Decomposition

The partial fraction decomposition for the fraction \( \frac{3x^2 - 7x + 33}{x^3 + 27} \) is set up by writing the fraction as a sum of simpler fractions. The denominator is now \((x + 3)(x^2 - 3x + 9)\). We write:\[\frac{3x^2 - 7x + 33}{x^3 + 27} = \frac{A}{x + 3} + \frac{Bx + C}{x^2 - 3x + 9}\]where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Clear the Denominator

Multiply through by the common denominator \((x+3)(x^2 - 3x + 9)\) to clear the fractions:\[3x^2 - 7x + 33 = A(x^2 - 3x + 9) + (Bx + C)(x + 3)\]This equation is now free of fractions.

Step 4: Expand and Collect Like Terms

Expand the right hand side of the equation:\[A(x^2 - 3x + 9) = Ax^2 - 3Ax + 9A\]\[(Bx + C)(x + 3) = Bx^2 + 3Bx + Cx + 3C\]Add these expanded terms together:\[(A + B)x^2 + (-3A + 3B + C)x + (9A + 3C)\]

Step 5: Set Up System of Equations

Now, equate the coefficients of the polynomial on both sides:1. Coefficients of \(x^2\): \( A + B = 3 \)2. Coefficients of \(x\): \(-3A + 3B + C = -7 \)3. Constant terms: \(9A + 3C = 33 \)

Step 6: Solve the System of Equations

Solve these equations:- From equation 1, solve for \(B\): \( B = 3 - A \)- Substitute \(B\) in equations 2 and 3: - \[-3A + 3(3 - A) + C = -7\] yields \(-6A + 9 + C = -7\) - Thus, \(C = -7 + 6A - 9 = 6A - 16\) - \[9A + 3(6A - 16) = 33\] simplifies to \(27A - 48 = 33\) - Solving gives \(A = 3\)- Substitute back to find \(B = 0, C = 2\).

Step 7: Write the Final Partial Fraction

Substituting \(A = 3\), \(B = 0\), and \(C = 2\) into the partial fraction decomposition gives:\[\frac{3x^2 - 7x + 33}{x^3 + 27} = \frac{3}{x+3} + \frac{2}{x^2 - 3x + 9}\]

Key Concepts

Irreducible Quadratic Factor

In algebra, a quadratic factor is termed 'irreducible' over the real numbers when it cannot be factored any further using real numbers. For the problem at hand, the expression

• \(x^2 - 3x + 9\)

represents an irreducible quadratic factor. This means there's no simple factorization using real numbers, as it doesn't cross the x-axis, implying no real roots. Instead, one can confirm its irreducibility by checking the discriminant. The discriminant for a quadratic \(ax^2 + bx + c\) is \(b^2 - 4ac\). If this is negative, the quadratic is irreducible. Here,

• \((-3)^2 - 4(1)(9) = 9 - 36 = -27\)

This negative result confirms the irreducibility.

Sum of Cubes Formula

The sum of cubes formula is crucial in factorizing certain polynomial expressions. For expressions of the form \(a^3 + b^3\), they can be factorized using the identity:

• \((a + b)(a^2 - ab + b^2)\)

This formula decomposes the cubic expression into two factors: a linear term and a quadratic term. In the problem, the denominator

• \(x^3 + 27 = x^3 + 3^3\)

was factorized into

• \((x + 3)(x^2 - 3x + 9)\)

Here, \(a\) is \(x\) and \(b\) is \(3\). Recognizing this structure facilitates the partial fraction decomposition by breaking complex polynomials into simpler terms.

System of Equations

A system of equations arises when you set up conditions to resolve unknown variables in a partial fraction decomposition. By clearing the denominator, we transform a polynomial identity into systems of equations based on matching coefficients. In the current problem, we're asked to solve for constants

• \(A, B,\) and \(C\)

Once the denominator's fraction is multiplied through and terms collected, we compare coefficients of similar powers of \(x\) on both sides. This generates a system of equations:

• \(A + B = 3\)
• \(-3A + 3B + C = -7\)
• \(9A + 3C = 33\)

Solving allows us to flesh out the specific values of the constants:

• \(A = 3, B = 0, C = 2\)

These values lead to the final expression of the partial fractions.

Polynomial Coefficients

Understanding polynomial coefficients is fundamental in solving polynomial identities and subsequently systems of equations. Coefficients are essentially the numerical coefficients prefixed to variable terms in polynomial expressions. During the process of polynomial decomposition, equating coefficients of terms of similar degree between two polynomial expressions is a common strategy. For instance, in the function given:

• \(3x^2 - 7x + 33 = A(x^2 - 3x + 9) + (Bx + C)(x + 3)\)

we equate the coefficients of like terms on both sides:

• \(x^2\)'s gives \(A + B = 3\)
• \(x\)'s gives \(-3A + 3B + C = -7\)
• constant terms gives \(9A + 3C = 33\)

By setting up such equations and resolving them, we solve for the necessary constants to express the original polynomial's partial fraction.