Question

problem 1. (1 point)
which of the following provides the best partial fraction decomposition for the function $f(x) = \dfrac{1}{(x - 6)^3(x^2 + 49)^2}$?
\\(\bigcirc\\) a. $\dfrac{ax^3 + b^2 + c}{(x - 6)^3} + \dfrac{dx^4 + ex^2 + f}{(x^2 + 49)^2}$
\\(\bigcirc\\) b. $\dfrac{a}{x - 6} + \dfrac{b}{(x - 6)^2} + \dfrac{c}{(x - 6)^3} + \dfrac{dx + e}{x^2 + 49} + \dfrac{fx + g}{(x^2 + 49)^2}$
\\(\bigcirc\\) c. $\dfrac{a}{x - 6} + \dfrac{b}{(x - 6)^2} + \dfrac{c}{(x - 6)^3} + \dfrac{d}{x^2 + 49} + \dfrac{e}{(x^2 + 49)^2}$
\\(\bigcirc\\) d. $\dfrac{a}{(x - 6)^3} + \dfrac{bx + c}{(x^2 + 49)^2}$
\\(\bigcirc\\) e. $\dfrac{a}{x - 6} + \dfrac{b}{(x - 6)^2} + \dfrac{c}{(x - 6)^3} + \dfrac{d}{x - 7} + \dfrac{e}{x + 7} + \dfrac{f}{(x - 7)^2} + \dfrac{g}{(x + 7)^2}$

Explanation:

To determine the partial fraction decomposition of \( f(x)=\frac{1}{(x - 6)^{3}(x^{2}+49)^{2}} \), we use the rules for partial fraction decomposition:

Step 1: Analyze the linear factor \((x - 6)^{3}\)

For a repeated linear factor \((ax + b)^n\), the partial fraction decomposition includes terms of the form \(\frac{A_1}{ax + b}+\frac{A_2}{(ax + b)^2}+\cdots+\frac{A_n}{(ax + b)^n}\). Here, \(ax + b=x - 6\) and \(n = 3\), so we have terms \(\frac{A}{x - 6}+\frac{B}{(x - 6)^2}+\frac{C}{(x - 6)^3}\).

Step 2: Analyze the irreducible quadratic factor \((x^{2}+49)^{2}\)

For a repeated irreducible quadratic factor \((ax^{2}+bx + c)^n\) (where \(ax^{2}+bx + c\) has no real roots), the partial fraction decomposition includes terms of the form \(\frac{B_1x + C_1}{ax^{2}+bx + c}+\frac{B_2x + C_2}{(ax^{2}+bx + c)^2}+\cdots+\frac{B_nx + C_n}{(ax^{2}+bx + c)^n}\). Here, \(ax^{2}+bx + c=x^{2}+49\) (which has discriminant \(0 - 4\times49=- 196<0\), so it's irreducible) and \(n = 2\), so we have terms \(\frac{Dx + E}{x^{2}+49}+\frac{Fx + G}{(x^{2}+49)^2}\).

Combining these two parts, the partial fraction decomposition of \(\frac{1}{(x - 6)^{3}(x^{2}+49)^{2}}\) is \(\frac{A}{x - 6}+\frac{B}{(x - 6)^2}+\frac{C}{(x - 6)^3}+\frac{Dx + E}{x^{2}+49}+\frac{Fx + G}{(x^{2}+49)^2}\), which matches option B.

Answer:

B. \(\frac{A}{x - 6}+\frac{B}{(x - 6)^{2}}+\frac{C}{(x - 6)^{3}}+\frac{Dx + E}{x^{2}+49}+\frac{Fx + G}{(x^{2}+49)^{2}}\)