How do you simplify and restricted value of #(x^3 - 2x + 3)/( x^2 + 12x + 32)#?

Question

Abigail Allen · Accepted Answer

#frac(x^(3) - 2 x + 3)((x + 6)^(2))#; #x ne - 6#

We have: #frac(x^(3) - 2 x + 3)(x^(2) + 12 x + 32)#

Let's factorise the denominator using the "middle-term break":

#= frac(x^(3) - 2 x + 3)(x^(2) + 6 x + 6 x + 32)#

#= frac(x^(3) - 2 x + 3)(x(x + 6) + 6(x + 6))#

#= frac(x^(3) - 2 x + 3)((x + 6)(x + 6))#

#= frac(x^(3) - 2 x + 3)((x + 6)^(2))#

Now let's determine the restricted values of #x#.

The denominator of the fraction can never equal to zero:

#Rightarrow (x + 6)^(2) ne 0#

#Rightarrow x + 6 ne 0#

#Rightarrow x ne - 6#

#therefore frac(x^(3) - 2 x + 3)(x^(2) + 12 x + 32) = frac(x^(3) - 2 x + 3)((x + 6)^(2))#; #x ne - 6#

Julia Adams · Answer

undefined To simplify the expression (x^3 - 2x + 3)/(x^2 + 12x + 32), we can factor the numerator and denominator. The numerator can be factored as (x - 1)(x^2 + x - 3), and the denominator can be factored as (x + 4)(x + 8).

Therefore, the expression can be simplified as (x - 1)(x^2 + x - 3)/(x + 4)(x + 8).

To find the restricted values, we need to identify the values of x that would make the denominator equal to zero. In this case, the restricted values are x = -4 and x = -8, as they would make the denominator (x + 4)(x + 8) equal to zero.