Question

solve the equation. remember to check for extraneous solutions.
\\(\frac{2}{p}=\frac{5p - 19}{p^{2}}\\)
\\(p=-\frac{17}{5}\\)
\\(p = \frac{19}{3}\\)
\\(p=-\frac{19}{7}\\)
\\(p = \frac{21}{5}\\)
question #3
simplify the rational expression
\\(\frac{4x^{3}+2x^{2}-5x}{x^{3}-x}\\)
\\(\frac{6x - 5}{x}\\)
\\(\frac{2x^{2}}{3x^{3}-6x}\\)
\\(\frac{2x - 5}{x - 1}\\)
\\(\frac{4x^{2}+2x - 5}{x^{2}-1}\\)

Explanation:

Step1: Solve the first - equation

Cross - multiply the equation $\frac{2}{p}=\frac{5p - 19}{p^{2}}$. Since $p
eq0$ (because if $p = 0$, the original equation is undefined), we get $2p^{2}=p(5p - 19)$. Expand the right - hand side: $2p^{2}=5p^{2}-19p$. Move all terms to one side: $5p^{2}-2p^{2}-19p = 0$, which simplifies to $3p^{2}-19p = 0$. Factor out $p$: $p(3p - 19)=0$. Set each factor equal to zero: $p = 0$ (extraneous, discard) or $3p-19 = 0$. Solving $3p-19 = 0$ gives $p=\frac{19}{3}$.

Step2: Simplify the rational expression

Factor the numerator and denominator of $\frac{4x^{3}+2x^{2}-5x}{x^{3}-x}$.
The numerator $4x^{3}+2x^{2}-5x=x(4x^{2}+2x - 5)$.
The denominator $x^{3}-x=x(x^{2}-1)=x(x + 1)(x - 1)$.
Cancel out the common factor $x$ (assuming $x
eq0$), and we get $\frac{4x^{2}+2x - 5}{x^{2}-1}$.

Answer:

First question: $p=\frac{19}{3}$
Second question: $\frac{4x^{2}+2x - 5}{x^{2}-1}$