Consider the two expressions and
a) Show that the two expressions are equal to each other when x=10
b) Explain why these two expressions are not equal to each other when x=-1/2
c) Show that these two expressions are equal to each other for all other than x=-1/2
In parts (a) and (c), begin by explaining what your strategy for solving will be.
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.
Part c: rewrite the second expression using the common denominator of 2x + 1
2x2 · [ (2x +1) / (2x + 1) ] + 3· [ (2x +1) / (2x + 1) ] + (4 )/ (2x + 1)
= (4x3 + 2x2 ) / (2x + 1) + (6x + 3) / (2x + 1) + (4) / (2x + 1)
= [ (4x3 + 2x2 ) + (6x + 3) + (4) ] / (2x + 1)
= ( 4x3 + 2x2 + 6x + 7 ) / (2x + 1)
Get a common denominator for the second expression, so the addition can be done above a single denominator.
( 2x2 + 3 )( 2x + 1) = 4x3 + 2x2 + 6x + 3
so the second expression can be rewritten as 4x3 + 2x2 + 6x + 3 4
———————— + ———
2x + 1 2x + 1
when you add those together, it is 4x3 + 2x2 + 6x + 3 + 4 4x3 + 2x2 + 6x + 7
—————————— and adding the 3 + 4 we obtain ————————
2x + 1 2x + 1
This is the same as the first expression, so the two are equivalent for every
value (not just 10) except x = –1/2 because when x = –1/2 the denominator
becomes zero and Rule No. 1 is "Thou shall not divide by zero."