Question

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 parenthesis, 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

Answer 1

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 $k = 2$