3. Division of Algebraic Expressions

First, we multiply out the top line:

`(12a^3b^6)/(8a^2b^3)`

When we write it out in full, this means

`(12 xx aaa xx b b b b b b)/(8 xx aa xx b b b)`

Next, cancel the numbers top and bottom (we divide top and bottom by `4`), the "a" terms (we cancel `a^2=aa` from top and bottom) and the "b" terms (we cancel `b^3=b b b` from top and bottom) to give us the final answer:

`(3ab^3)/2`

We square the denominator (bottom) of the fraction:

`(12m^2n^3)/((6m^4n^5)^2)=(12m^2n^3)/(36m^8n^10)`

Next, we cancel out the numbers, and the "m" and "n" terms to give the final answer:

`1/(3m^6n^7)`

With this example, we'll break it into 2 fractions, both with denominator 4q to make it easier to see what to do.

`(6p^3q^2-10p^2q)/(4q)=(6p^3q^2)/(4q)-(10p^2q)/(4q)`

Next, we cancel the numbers and variables:

`(3p^3q)/2-(5p^2)/2`

Finally, we combine the fractions:

`(3p^3q-5p^2)/2`

After you have had some practice with these, you'll be able to do it without separating them into 2 fractions first.

I'll show you how to do this two different ways. It is worth seeing both, because they are both useful. You can decide which is easier ;-)

Solution 1 - Multiplying by the Reciprocal

I take the top expression (numerator) and turn it into a single fraction with denominator x.

`3+1/x=(3x+1)/x`

We do likewise with the bottom expression (denominator):

`5/x+4=(5+4x)/x`

So the question has become:

`(3+1/x)/(5/x+4)=((3x+1)/x)/((5+4x)/x)`

We think of the right side as a division of the top by the bottom:

`(3x+1)/x-:(5+4x)/x`

To divide by a fraction, you multiply by the reciprocal:

`(3x+1)/(x)xxx/(5+4x)=(3x+1)/(5+4x)`

The x's cancelled out, and we have our final answer, which cannot be simplified any more.

Solution 2 - Multiplying Top and Bottom

I recognise that I have "/x" in both the numerator and denominator. So if I just multiply top and bottom by x, it will simplify everything by removing the fractions on top and bottom.

`(3+1/x)/(5/x+4)xxx/x`

I am really just multiplying by "1" and not changing the original value of the fraction - just changing its form.

So I multiply each element of the top by x and each element of the bottom by x and I get:

`(3+1/x)/(5/x+4)xxx/x=(3x+1)/(5+4x)`

I cannot simplify any further.

We are dividing a polynomial of degree 2 by a polynomial of degree 1. This is algebraic long division.

Step 1: We look at the first term of (3x² − 11x − 4) and the first term of (x − 4).

Divide as follows: 3x² ÷ x = 3x

We write 3x at top of our long division and multiply (3x)(x − 4) = 3x² − 12x to give the second row of our solution.

Step 2: Subtracting the second row from the first gives:

Be careful with

-11x − (-12x) = -11x + 12x = x

Step 3: Bring down the -4 from the first row:

Step 4: Divide x (in the 3rd row) by x from the (x − 4) in the question. Our answer is 1 and we write "+ 1" at the top of our long division.

Next, multiply (1) by (x − 4) to get the 4th row.

Step 5: Subtract the 4th row from the 3rd:

So (3x² − 11x − 4) ÷ (x − 4) = 3x + 1

You can check your answer by multiplying (3x + 1) by (x − 4) and you'll get (3x² − 11x − 4).

We can think of `(6x^2+6+7x)/(2x+1)` as (6x² + 7x + 6) ÷ (2x + 1)

Once again we are dividing a polynomial of degree 2 by a polynomial of lower degree (1). This is algebraic long division.

Step 1: 6x² ÷ 2x = 3x

So we write the following, using (3x)(2x + 1) = 6x² + 3x for the second row:

Step 2: We subtract 6x² + 3x from the first row:

Step 3: Bring down the 6:

Step 4: Divide 4x by 2x. Our answer is 2 and we multiply 2(2x + 1) to get the 4th row.

Step 5: Subtract, and we are left with 4.

So the answer is:

`(6x^2+6+7x)/(2x+1)=3x+2+4/(2x+1)`

NOTE: Some people prefer to write the problem with all the x²'s, x's and units in line, as follows: