# How do you find the excluded value and simplify  (x^2-13x+42)/(x+7)?

May 12, 2018

$\text{excluded value } = - 7$

#### Explanation:

The denominator of the rational expression cannot be zero as this would make it undefined. Equating the denominator to zero and solving gives the value that x cannot be.

$\text{solve "x+7=0rArrx=-7larrcolor(red)"excluded value}$

$\text{to simplify factorise the numerator and cancel any }$
$\text{common factors}$

$\text{the factors of + 42 which sum to - 13 are - 6 and - 7}$

$\Rightarrow {x}^{2} - 13 x + 42 = \left(x - 6\right) \left(x - 7\right)$

$\Rightarrow \frac{{x}^{2} - 13 x + 42}{x + 7}$

$= \frac{\left(x - 6\right) \left(x - 7\right)}{x + 7} \leftarrow \textcolor{red}{\text{in simplest form}}$

May 12, 2018

Restriction: $x \setminus \ne - 7$ , simplified expression: Already simplified

#### Explanation:

since the denominator is $x + 7$ and you cannot divide by zero, $x + 7 \setminus \ne 0$ thus, $x \setminus \ne - 7$
next because the expression on the numerator is a quadratic, it can probably be factored. All that is needed is two numbers that add up to -13 ad two numbers that multiply to 42.

If you factor 42 you get: $\setminus \pm \left[1 , 2 , 3 , 6 , 7 , 14 , 21 , 42\right]$
notice that -6 and -7 add up to -13 and multiply to 42 thus:

${x}^{2} - 13 x + 42 = {x}^{2} - 6 x - 7 x + 42 = x \left(x - 6\right) - 7 \left(x - 6\right) = \left(x - 6\right) \left(x - 7\right)$

None of these linear factors cancel out with the denominator and thus the expression cannot be simplified.