Long Division
Main Topic # 1: [Long Division with polynomials] The basic concept of worksheet is write rational expressions
as as a polynomial plus a simpler rational expression. This idea was seen with numbers when you were younger.
More specifically we can find a qcalled the quotient and rthe remainder so that
a
b=q+r
b
so that r < b. This is not as tricky as it might seem to the mathematical adverse student. It is only a tricky way
of writing that there is a number qsuch that
a=q·b+r
that is qis the most amount of times bdivides into aan ris the left over, that is why it is less than bif it was
greater we could get another factor of b. Now we want to do the same process with polynomials specifically lets
look at
(x2+ 2x−1)
| {z }
dividend
÷(x−1)
| {z }
divisor
We begin very much the same as long division of numbers but this time we only look at the highest term in each.
The highest term in x−1 is just x1and the highest term in x2+ 2x−1 is x2. We even write the set up the same!
x−1x2+ 2x−1
So we first ask the question what can we multiply x(i.e. our highest term in our divisor) to get x2(i.e. our highest
term in our dividend) well that is of course xor to put it in an equation
x·x=x2
We collect this data just as before by writing it above the line
x
x−1x2+ 2x−1
We then proceed like with numbers and multiply our new factor xby (x−1) and get
x·(x−1) = x2−x
and again we must subtract (don’t forget to distribute the subtraction sign i.e. −(x2−x) = −x2+x
x
x−1x2+ 2x−1
−x2+x
and then performing the subtraction by subtracting like terms we get
x
x−1x2+ 2x−1
−x2+x
3x−1
Now we repeat the process with the left over polynomial in this example it is 3x−1. Notice the highest term of
3x−1 is 3xand we ask what do we need to multiply x(the highest term of x−1) by to get 3x? We see here it is
more simple we need only multiply by 3! That is
3·x= 3x
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