Question

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

Answer 1

`"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.

`"solve "x+7=0rArrx=-7larrcolor(red)"excluded value"`

`"to simplify factorise the numerator and cancel any "`
`"common factors"`

`"the factors of + 42 which sum to - 13 are - 6 and - 7"`

`rArrx^2-13x+42=(x-6)(x-7)`

`rArr(x^2-13x+42)/(x+7)`

`=((x-6)(x-7))/(x+7)larrcolor(red)"in simplest form"`

Answer 2

Restriction: `x \ne -7` , simplified expression: Already simplified

Explanation:

since the denominator is `x+7` and you cannot divide by zero, `x+7 \ne 0` thus, `x \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: `\pm[1,2,3,6,7,14,21,42]`
notice that -6 and -7 add up to -13 and multiply to 42 thus:

`x^2-13x+42 = x^2-6x-7x+42 = x(x-6) -7(x-6) = (x-6)(x-7)`

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