Before taking up division of polynomials, let us acquaint ourselves with some basics. Suppose we are asked to divide 16 by 2. We know that on dividing 16 by 2 we get 8. In mathematics we call 16, 2 and 8 by specific names. 16 is called dividend, 2 is called the divisor and 8 is the quotient. However, it is not always that we get an integer like 8 when we divide a number by another. For instance, divide 9 by 2. In addition to the dividend (9), divisor (2) and quotient (4) we are left with another term 1. This is referred to as the remainder. When the dividend is not exactly divisible by the divisor we get a remainder. We find these terms even when one expression is divided by another. Also we follow these rules.
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We arrange the terms of the divisor and the dividend in ascending or descending powers of some common letter. Ascending order refers to arranging terms from lower power to higher powers and descending orders refers to opposite of this. Usually we write them in the descending order.
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Divide the term on the left of the dividend by the term left of the divisor and put the result in the quotient.
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Multiply the whole divisor by this number (quotient) and put the resultant product under the dividend.
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Subtract the product from the dividend and bring down the required number of terms as may be deemed necessary.
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Repeat this procedure until all the terms in the dividend have been brought down.
We understand this with the help of a couple of examples.
Example
Divide x2 + 4x + 4 by x + 2.
We find that the terms of the dividend (x2 + 4x + 4) and the divisor (x + 2) are already in the descending order. The left most term in the dividend is x2, while in the divisor it is x. We find the quotient as
We multiply the divisor x + 2 with this quotient x. We get x2 + 2x. We write this under the dividend as shown.
Others
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x + 2 )
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x2 + 4x + 4
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( x + 2
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(-)
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x2 + 2x
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2x + 4
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(-) 2x + 4
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0
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On subtracting x2 + 2x from the dividend we obtain 2x + 4. (x2 + 4x + 4 - (x2 + 2x) = x2 + 4x + 4 - x2 - 2x). We write this expression as shown above.
At this stage, we take the left most quantity of the difference (dividend - product) and that of the divisor and obtain their quotient. It will be
Since the sign of the quotient is positive we write it as shown. Then we multiply x + 2 with 2. That will be 2x + 4. We write under the difference got earlier and subtract it from the difference. We get 2x + 4 - (2x + 4) = 2x + 4 - 2x - 4 = 0. This is shown in the example above. Since the dividend is exactly divisible by the divisor the remainder is zero.
After solving this problem can we say that x + 2 is a factor of x2 + 4x + 4? Of course we can. As we write 8 = 2.4 or 1.8, we can write
x2 + 4x + 4 = (x + 2)(x + 2)
(Note: Division of expressions where some of the terms are fractions is also carried out in the same manner we have seen above.)