Question

I have the answer, 2-(3/x+2)

i got this from my calculator, however i need another line of working and i’m unsure of the process used to get there

Answer 1

a = 2

b = -3

Explanation:

the secret is seeing that the numerator (top part of the division) contains 2x. that means 2 times the factor of x in the denominator (bottom part of the division).

so, we want to change the numerator that we can simply say the result is 2 and some rest (remainder).

2×(x+2) would be 2x + 4

aha !

and we have 2x+1 up there. so, what had to happen to get from 2x+4 to 2x+1 ? we had to subtract 3. it to get to 2x+4 we have to add 3.

but if we add 3, we need also to subtract 3 to keep the value of the whole expression the same.

therefore we get

(2x+1)/(x+2) = (2x+4)/(x+2) - 3/(x+2) =

= 2×(x+2)/(x+2) - 3/(x+2) = 2 - 3/(x+2)

Answer 2

Start with the answer format we want, and work your way toward forming a single fraction like so

`a + (b)/(x+2)\\\\a*1+(b)/(x+2)\\\\a*(x+2)/(x+2)+(b)/(x+2)\\\\(a(x+2))/(x+2)+(b)/(x+2)\\\\(a(x+2)+b)/(x+2)\\\\(ax+2a+b)/(x+2)\\\\(ax+(2a+b))/(x+2)\\\\`

Compare that last expression to (2x+1)/(x+2). Notice how the ax and 2x match up, so a = 2 must be the case.

Then we have 2a+b as the remaining portion in the numerator. Plugging in a = 2 leads to 2a+b = 2*2+b = 4+b. Set this equal to the +1 found in (2x+1)/(x+2) to have the terms match.

So, 4+b = 1 leads to b = -3

Therefore, a = 2 and b = -3

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An alternative route:

`(2x+1)/(x+2)\\\\(2x+1+0)/(x+2)\\\\(2x+1+4-4)/(x+2)\\\\((2x+4)+1-4)/(x+2)\\\\(2(x+2)-3)/(x+2)\\\\(2(x+2))/(x+2)+(-3)/(x+2)\\\\2-(3)/(x+2)\\\\`

I added and subtracted 4 in the third step so that I could form 2x+4, which then factors to 2(x+2). That way I could cancel out a pair of (x+2) terms toward the very end.

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Other alternative methods involve synthetic division or polynomial long division. They are slightly separate but related concepts.