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

2 Answers
May 12, 2018

Answer:

#"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"#

May 12, 2018

Answer:

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.