Question

item 5 simplify rational expressions
retry 1 of 2
simplify the rational expression.
\\(\frac{c^2 + 5c - 24}{c^2 - 6c + 9}\\)
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simplify the rational expression.
\\(\frac{w^2 + \frac{10}{21}w + \frac{1}{21}}{w^2 - \frac{2}{35}w - \frac{1}{35}}\\)
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Explanation:

First Rational Expression: $\boldsymbol{\frac{c^2 + 5c - 24}{c^2 - 6c + 9}}$

Step1: Factor Numerator

We factor the quadratic $c^2 + 5c - 24$. We need two numbers that multiply to $-24$ and add to $5$. The numbers are $8$ and $-3$. So, $c^2 + 5c - 24=(c + 8)(c - 3)$.

Step2: Factor Denominator

We factor the quadratic $c^2 - 6c + 9$. This is a perfect square trinomial, since $(-3)^2 = 9$ and $2\times c\times(-3)=-6c$. So, $c^2 - 6c + 9=(c - 3)^2$.

Step3: Simplify the Expression

Now we substitute the factored forms into the rational expression: $\frac{(c + 8)(c - 3)}{(c - 3)^2}$. We can cancel out one $(c - 3)$ term from the numerator and the denominator (assuming $c
eq3$ to avoid division by zero). This gives us $\frac{c + 8}{c - 3}$.

Step1: Eliminate Fractions (Numerator)

Multiply the numerator $w^2+\frac{10}{21}w+\frac{1}{21}$ by $21$ to eliminate fractions: $21w^2 + 10w + 1$. Now factor this quadratic. We need two numbers that multiply to $21\times1 = 21$ and add to $10$. The numbers are $7$ and $3$. So, $21w^2 + 10w + 1 = 21w^2+7w + 3w + 1=7w(3w + 1)+1(3w + 1)=(7w + 1)(3w + 1)$.

Step2: Eliminate Fractions (Denominator)

Multiply the denominator $w^2-\frac{2}{35}w-\frac{1}{35}$ by $35$ to eliminate fractions: $35w^2 - 2w - 1$. Now factor this quadratic. We need two numbers that multiply to $35\times(-1)=-35$ and add to $-2$. The numbers are $-7$ and $5$. So, $35w^2 - 2w - 1=35w^2+5w - 7w - 1 = 5w(7w + 1)-1(7w + 1)=(5w - 1)(7w + 1)$.

Step3: Simplify the Expression

Now we have the rational expression (after multiplying numerator and denominator by their respective least common multiples, which is equivalent to multiplying by $\frac{21\times35}{21\times35}=1$ so the value remains the same) as $\frac{(7w + 1)(3w + 1)}{(5w - 1)(7w + 1)}$. We can cancel out the $(7w + 1)$ term from the numerator and the denominator (assuming $7w+1
eq0$ to avoid division by zero). This gives us $\frac{3w + 1}{5w - 1}$.

Answer:

$\frac{c + 8}{c - 3}$

Second Rational Expression: $\boldsymbol{\frac{w^2+\frac{10}{21}w+\frac{1}{21}}{w^2-\frac{2}{35}w-\frac{1}{35}}}$