How do you simplify and find the restrictions for #(2x^2+11x+5)/(3x^2+17x+10)#?

Question

Isaiah Anthony · Accepted Answer

See a solution process below:

First, factor the numerator and denominator: #(2x^2 + 11x + 5)/(3x^2 + 17x + 10) => ((2x + 1)(x + 5))/((3x + 2)(x + 5))# Now, cancel common terms in the numerator and denominator: #((2x + 1)color(red)(cancel(color(black)((x + 5)))))/((3x + 2)color(red)(cancel(color(black)((x + 5))))) => (2x + 1)/(3x + 2)# To find the restrictions the denominator cannot be #0# therefore we need to solve for: #3x^2 + 17x + 10 =# Or #(3x + 2)(x + 5) = 0# Solution 1) #3x + 2 = 0# #3x + 2 - color(red)(2) = 0 - color(red)(2)# #3x + 0 = -2# #3x = -2# #(3x)/color(red)(3) = -2/color(red)(3)# #(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) = -2/3# #x = -2/3# Solution 2) #x + 5 = 0# #x + 5 - color(red)(5) = 0 - color(red)(5)# #x + 0 = -5# #x = -5# Therefore, the restrictions are: #x != -2/3# and #x != -5#

Isaiah Anthony · Answer

undefined To simplify the expression (2x^2+11x+5)/(3x^2+17x+10), you can factor both the numerator and denominator.

The numerator can be factored as (2x+1)(x+5), and the denominator can be factored as (3x+2)(x+5).

Now, you can cancel out the common factors of (x+5) from both the numerator and denominator.

After canceling out the common factors, the simplified expression becomes (2x+1)/(3x+2).

To find the restrictions, you need to identify the values of x that would make the denominator equal to zero.

In this case, the denominator (3x+2) would be equal to zero when x = -2/3.

Therefore, the restriction for this expression is x ≠ -2/3.