Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

What is the sum?

[tex]\[ \frac{3y}{y^2 + 7y + 10} + \frac{2}{y + 2} \][/tex]

A. \(\frac{5}{y-5}\)

B. \(\frac{5(y+2)}{(y-2)(y+5)}\)

C. \(\frac{5}{y+5}\)

D. [tex]\(\frac{5(y-2)}{(y-5)(y+2)}\)[/tex]


Sagot :

To find the sum of the given fractions, we first need to factorize the denominators where possible and then combine the fractions into a single rational expression before simplifying. Let's walk through this step-by-step.

The given expression is:
[tex]\[ \frac{3 y}{y^2+7 y+10}+\frac{2}{y+2} \][/tex]

First, let's factorize the quadratic expression in the denominator \(y^2 + 7y + 10\). We look for two numbers that multiply to 10 (the constant term) and add up to 7 (the coefficient of the linear term \(y\)):

[tex]\[ y^2 + 7y + 10 = (y + 2)(y + 5) \][/tex]

So, the expression now is:
[tex]\[ \frac{3 y}{(y + 2)(y + 5)} + \frac{2}{y + 2} \][/tex]

Next, we need a common denominator to combine these fractions. The least common denominator (LCD) will be \((y + 2)(y + 5)\).

Rewrite each fraction with the common denominator:
[tex]\[ \frac{3 y}{(y + 2)(y + 5)} + \frac{2(y + 5)}{(y + 2)(y + 5)} \][/tex]

Now, combine the numerators over the common denominator:
[tex]\[ \frac{3y + 2(y + 5)}{(y + 2)(y + 5)} \][/tex]

Distribute and simplify the numerator:
[tex]\[ 3y + 2(y + 5) = 3y + 2y + 10 = 5y + 10 \][/tex]

So our fraction becomes:
[tex]\[ \frac{5y + 10}{(y + 2)(y + 5)} \][/tex]

Observe that the numerator \(5y + 10\) can be factored out:
[tex]\[ 5y + 10 = 5(y + 2) \][/tex]

Thus, the fraction simplifies to:
[tex]\[ \frac{5(y + 2)}{(y + 2)(y + 5)} \][/tex]

The \((y + 2)\) terms in the numerator and denominator cancel out:
[tex]\[ \frac{5}{y + 5} \][/tex]

Therefore, the simplified sum of the given fractions is:
[tex]\[ \frac{5}{y + 5} \][/tex]

The correct answer is:
[tex]\[ \boxed{\frac{5}{y+5}} \][/tex]