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Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two natural numbers is the process of calculating the number of times one number is contained within one another. For example, the 20 apples are divided into groups of five apples, and there exist four groups, meaning that five can be contained within 20 four times, or 20 ÷ 5 = 4. Division can also be thought of as the process of evaluating a fraction, and fractional notation (a/b and a⁄b) is commonly used to represent division. Division by zero is undefined for the real numbers and most other contexts.

Division of Fractions:

Three simple steps of dividing fractions are:

Step 1. Turn the second fraction (the one you want to divide by) upside down
(this is now a reciprocal).

Step 2. Multiply the first fraction by that reciprocal.

Step 3. Simplify the fraction (if needed).

Example:

÷

Step 1. Turn the second fraction upside down (it becomes a reciprocal):

becomes

Step 2. Multiply the first fraction by that reciprocal:

(multiply tops ...)

×    =

(... multiply bottoms)

Division of Polynomials:

f you're dividing a polynomial by something more complicated than just a simple monomial, then you'll need to use a different method for the simplification. That method is called "long (polynomial) division", and it works just like the long (numerical) division you did back in elementary school, except that now you're dividing with variables.

For example, Divide x2 – 9x – 10 by x + 1

 First, I set up the division: For the moment, I'll ignore the other terms and look just at the leading x of the divisor and the leading x2 of the dividend. If I divide the leading x2 inside by the leading x in front, what would I get? I'd get an x. So I'll put an x on top: Now I'll take that x, and multiply it through the divisor, x + 1. First, I multiply the x (on top) by the x (on the "side"), and carry the x2 underneath: Then I'll multiply the x (on top) by the 1 (on the "side"), and carry the 1x underneath: Then I'll draw the "equals" bar, so I can do the subtraction. To subtract the polynomials, I change all the signs in the second line... ...and then I add down. The first term (the x2) will cancel out: I need to remember to carry down that last term, the "subtract ten", from the dividend: Now I look at the x from the divisor and the new leading term, the –10x, in the bottom line of the division. If I divide the –10x by the x, I would end up with a –10, so I'll put that on top: Now I'll multiply the –10 (on top) by the leading x (on the "side"), and carry the –10x to the bottom: ...and I'll multiply the –10 (on top) by the 1 (on the "side"), and carry the –10 to the bottom: I draw the equals bar, and change the signs on all the terms in the bottom row: Then I add down:

Then the solution to this division is: x – 10

Division with Decimals:

To divide decimal numbers:

1. If the divisor is not a whole number, move decimal point to right to make it a whole number and move decimal point in dividend the same number of places.

2. Divide as usual. Keep dividing until the answer terminates or repeats.

3. Put decimal point directly above decimal point in the dividend.

Example: Divide 9.1 by 7

Ignore the decimal point and use Long Division:

 13 7 )91    7    21    21     0

Put the decimal point in the answer directly above the decimal point in the dividend:

 1.3 7 )9.1

Example: Divide 6.4 by 0.4

We are not dividing by a whole number, so we need to move the decimal point so we are dividing by a whole number:

 move 1 6.4 64 0.4 4 move 1

6.4/0.4 is exactly the same as 64/4, as we moved the decimal point of both numbers.

Now we can calculate:

64 / 4 = 16

6.4 / 0.4 = 16

Division Worksheet:

Name: ………………………………….. Score:………………………………….

Teacher: ……………………………….. Date: ………………………………….

 58.8/999.6 62.7/4451.7 11.7/1041.3 75.3/5195.7 92.1/8565.3 47.9/4742.1 67.3/1480.6 85.4/4526.2 74.7/2913.3

Name: ………………………………….. Score:………………………………….

Teacher: ……………………………….. Date: ………………………………….

 9/864 5/495 2/44 9/855 6/528 4/340 7/672 6/354 5/370

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