Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression. $$(4x^3$$ + $$5x^2 + 7x$$ - $$1)/(x^2 + x$$ + $$1)^2$$

Accepted Answer

Step 1: Recognize that the given rational expression is a proper fraction, as the degree of the numerator $$(4x^3$$ + $$5x^2 + 7x - 1) is $$less than the degree of the denominator $$((x^2 + x$$ + $$1)^2). $$This means partial fraction decomposition is applicable. Step 2: Factor the denominator if possible. In this case, the denominator is already expressed as $$(x^2 + x$$ + $$1)^2$$, which is a repeated irreducible quadratic factor. Step 3: Set up the partial fraction decomposition. For a repeated irreducible quadratic factor like $$(x^2 + x$$ + $$1)^2$$, the decomposition will take the form: A(x) / $$(x^2 + x + 1) + B(x$$) / $$(x^2 + x$$ + $$1)^2$$, where A(x) and B(x) are polynomials of degree less than the degree of the quadratic factor (degree \u003c 2). Thus, A(x) = Ax + B and B(x) = Cx + D. Step 4: Write the equation for the decomposition: $$(4x^3$$ + $$5x^2 + 7x - 1$$) / $$(x^2 + x$$ + $$1)^2 = (Ax + B$$) / $$(x^2 + x + 1) + (Cx + D$$) / $$(x^2 + x$$ + $$1)^2. $$Step 5: Multiply through by the denominator $$(x^2 + x$$ + $$1)^2 to $$eliminate the fractions, resulting in: $$4x^3$$ + $$5x^2 + 7x - 1 = (Ax$$ + $$B)(x^2 + x + 1) + (Cx + D). $$Expand and collect like terms, then equate coefficients of corresponding powers of x to solve for A, B, C, and D.

In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^3 + 5x^2 + 7x - 1)/(x^2 + x + 1)^2

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