Consider the expressions \(\frac{4x^3+2x^2+6x+7}{2x+1}\) and \(2x^2+3+\frac4{2x+1}\)
a) Show that these expressions are equal when x=10
b) Explain why these expressions are not equal when x=-1/2
c) Show that these expressions are equal for all x other than -1/2
In parts (a) and (c), begin by explaining what your strategy for solving will be.
IMPORTANT:
Your first instinct on Part (c) may be to manipulate an equation until both sides are equal. However, this can confuse your reader: you would be writing equations that you don't know are true! Try to write your solution so that every equation you write is true
+128407 Here's part (c)
2x^2 + 3 + [ 4 ] / (2x + 1) can be simplified as
[ (2x^2 + 3)(2x + 1) + 4 ] / [ 2x + 1] =
[ 4x^3 + 2x^2 + 6x + 7 ] / [ 2x + 1]
Which is equal to the first function
With this in mind....(a) must be true
+63 Part b)
When x=-1/2. the denominator of the first expression becomes undefined. When x=-1/2, the second expression's third term becomes undefined too. Therefore, they are not equal because the first expression is entirely undefined while the second expression has only one term undefined.
