Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression. $$(7x^2 - 7x + 23)/(x$$ - $$3)(x^2 + 4)$$

Accepted Answer

Step 1: Recognize that the given rational expression $$(7x^2 - 7x + 23)/((x - 3)(x^2 + 4))$$ needs to be decomposed into partial fractions. The denominator consists of two distinct factors: $$x - 3 (a $$linear factor) and $$x^2 + 4 (an $$irreducible quadratic factor). Step 2: Set up the partial fraction decomposition. For the linear factor $$x - 3$$, assign a constant $$A as $$the numerator. For the irreducible quadratic factor $$x^2 + 4$$, assign a linear numerator $$Bx + C. $$The decomposition will look like: $$\frac{7x^2 - 7x + 23}{(x - 3)(x^2 + 4)} = \frac{A}{x - 3} + \frac{Bx + C}{x^2 + 4}. $$Step 3: Multiply through by the denominator $$(x - 3)(x^2 + 4) to $$eliminate the fractions. This gives: $$7x^2 - 7x + 23 = A(x^2 + 4) + (Bx + C)(x - 3). $$Step 4: Expand the right-hand side of the equation. Distribute $$A$$ across $$x^2 + 4$$, and distribute $$Bx + C$$ across $$x - 3. $$This results in: $$7x^2 - 7x + 23 = A(x^2) + A(4) + Bx(x) + Bx(-3) + C(x) + C(-3). $$Combine like terms. Step 5: Equate coefficients of $$x^2$$, $$x$$, and the constant terms on both sides of the equation. Solve the resulting system of equations to find the values of $$A$$, $$B$$, and $$C. $$Substitute these values back into the partial fraction decomposition.

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (7x^2 - 7x + 23)/(x - 3)(x^2 + 4)

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Key Concepts

Rational Expressions

Partial Fraction Decomposition

Polynomial Factorization