How do you simplify #(4g^2 - 64g + 252)/(g-7)#?

Question

Eva Adamson · Accepted Answer

Let's Factorise the #color(red)(NUMERATOR)# first. The Numerator is #color(red)(4g^2 - 64g + 252)# # = 4(g^2 - 16g + 63)#    (4 was the factor common to all terms) Now we need to factorise #color(blue) (g^2 - 16g + 63) # 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=-16, c= 63# 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)*(63)  and N_1+N_2 = 16#
#N_1*N_2 = 63  and N_1+N_2 = -16# After Trial and Error, we get #N_1 = -7 and N_2 = -9#
#(-7)*(-9) = 63# and #(-7) + (-9) = -16# So we can write the expression in blue as 
#color(blue) (g^2 - 7g -9g+ 63) #
#=g(g-7)-9(g-7)#
#= (g-7)*(g-9)# The Numerator can be written as #color(red)(4(g-7)*(g-9))# The expression we have been given is 
#(4g^2 - 64g + 252)/(g-7)# Following factorization of the numerator, the expression can now be expressed as follows: #(4(g-7)*(g-9))/(g-7)# # =(4*cancel((g-7))*(g-9))/cancel((g-7))# # = 4*(g-9) # #(4g^2 - 64g + 252)/(g-7)# = # 4*(g-9) #

Kennedy Adams · Answer

undefined To simplify the expression (4g^2 - 64g + 252)/(g-7), you can use polynomial long division or synthetic division. However, in this case, we can factor the numerator and cancel out common factors with the denominator.

The numerator can be factored as (2g - 6)(2g - 42).

Now, we can cancel out the common factor of (g - 7) from the numerator and denominator.

Therefore, the simplified expression is (2g - 6)(2g - 42)/(g - 7).