How do you simplify #(r+6)/(4^2-r-6)# and state the restrictions?

Question

Hazel Anglin · Accepted Answer

I assume your question is #(r + 6)/(r^2 - r - 6)# Let's Factorise the #color(red)(DENOMINATOR)# first. The Denominator is #color(red)(r^2 - r - 6)# To factorize this, we can apply the Splitting the Middle Term technique. It is in the form #ax^2 + bx + c# where #a=1, b=-1, c= -6# To split the middle term, we need to think of two numbers #N_1 and N_2# such that:
#N_1*N_2 = a*c  and N_1+N_2 = b#
#N_1*N_2 = (1)*(-6)  and N_1+N_2 = -1#
#N_1*N_2 = -6  and N_1+N_2 = -1# After Trial and Error, we get #N_1 = 2 and N_2 = -3#
#(2)*(-3) = -6# and #(2) + (-3) = -1# So we can write the denominator as 
#color(red)(r^2 +2r -3r - 6)#
# = r*(r+2) - 3*(r+2)#
# = (r+2)*(r-3)# The Denominator can be written as #color(red)((r-2)*(r+3))# The expression we have been given is 
#(r + 6)/(r^2 - r - 6)# Following factorization of the denominator, the expression can now be expressed as follows: #((r+6))/((r+2)*(r-3))# #(r + 6)/(r^2 - r - 6)# = #((r+6))/((r+2)*(r-3))#

Kennedy Adams · Answer

undefined To simplify (r+6)/(4^2-r-6), we first factor the denominator: (r+6)/((4-3)(4+2)). This simplifies to (r+6)/(1(6)), which further simplifies to (r+6)/6. The restrictions for this expression are that r cannot equal 4 or -2, as these values would result in division by zero.