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# How to do this

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Consider the two expressions $$\frac{6x^3+9x^2+4x+7}{2x+3}$$and $$3x^2+2+\frac1{2x+3}$$
a) Show that the two expressions represent equal numbers when  x=10

b) Explain why these two expressions do not represent equal numbers when $$x=-\dfrac32$$

c) Show that these two expressions represent equal numbers for all  x other than $$-\dfrac32$$
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.
Now when I simplify the two expression, the first one is 6x^3+9x^2+3. And the second one is 6x^3+9x^2+4, so how do I prove that the two expressions represent equal numbers when x=10.

Jul 16, 2023

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a)  if x = 10 we can just plug this into the equation :

(6 ( 10 ) ^ 3  +. 9 (10) ^ 2 + 4 ( 10 ) + 7 ).  /  2( 10 ) + 3    = 3 ( 10 ) ^ 2 + 2 + 1/ (2 (10) + 3)

(6000 + 900 + 40 + 7 ) / 23  = 300 + 2 +  1 / (23)

(6947) / 23 = 300 + 2 +  1/ 23

302  + (remainder of 1 ) = 302 + 1/23

We can divide the remainder of 1 by 23 since we are dividing by 23 thus :

302 + 1/23 = 302 = 1/23

I am confused about the ending paragraph however I think you can do it for the last 2 sections Jul 17, 2023