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olya-2409 [2.1K]
3 years ago
10

Divide the polynomials. Your answer should be in the form p(x)+\dfrac{k}{x+2}p(x)+ x+2 k ​ p, left parenthesis, x, right parenth

esis, plus, start fraction, k, divided by, x, plus, 2, end fraction where ppp is a polynomial and kkk is an integer. \dfrac{x^2+7x+12}{x+2}= x+2 x 2 +7x+12 ​ =start fraction, x, squared, plus, 7, x, plus, 12, divided by, x, plus, 2, end fraction, equals
Mathematics
1 answer:
pav-90 [236]3 years ago
6 0

Answer:

(x+5)+ \dfrac{2}{x+2}

Step-by-step explanation:

Given

\dfrac{x^2+7x+12}{x+2}

Required

Simplify and present your answer in

p(x)+\dfrac{k}{x+2}

First, we need to expand and factorize the numerator:

\dfrac{x^2+7x+12}{x+2}

Express 12 as 10 + 2

\dfrac{x^2+7x+10+2}{x+2}

Split into two using brackets

\dfrac{(x^2+7x+10)+2}{x+2}

Split to two fractions:

\dfrac{(x^2+7x+10)}{x+2}+\dfrac{2}{x+2}

Factorize the numerator of the first fraction

\dfrac{(x^2+5x+2x+10)}{x+2}+\dfrac{2}{x+2}

\dfrac{x(x+5)+2(x+5)}{x+2}+\dfrac{2}{x+2}

\dfrac{(x+2)(x+5)}{x+2}+\dfrac{2}{x+2}

The first fraction can be simplified to give:

(x+5)+ \dfrac{2}{x+2}

Hence, by comparison:

We have simplified the given expression to the required format of:

p(x)+\dfrac{k}{x+2}

Which implies that:

p(x) = x + 5 and k = 2

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