Dividing Polynomials is an algebraic operation to divide one polynomial with another. This follows the basic rule in fractions, whereby such a fraction can be rewritten as the sum of fractions consisting of each numerator term divided by the term in the denominator.
Basic Definitions:
- Numerator: The number above the line in a fraction is called a numerator. For example, 3/4 here 3 is called a numerator.
- Denominator: The number below the line is called denominator. For example, 3/4 here 4 is called denominator.
- Quotient: The result of the division is called Quotient. If you divide 12/4 you get the answer as 3, which is a quotient.
- Remainder: Some numbers are not completely divisible. There is left over after division which is called Remainder. Dividing 12/5 gives Quotient 2 and the remainder also as 2.
Dividing Polynomials By A Monomial:
Let’s first consider how to divide a polynomial by a monomial. In this type of division, the only thing necessary is to divide each term of the polynomial by the monomial. This follows the basic rule in fractions, whereby such a fraction can be rewritten as the sum of fractions consisting of
each numerator term divided by the term in the denominator.
$$ \frac{a\;+\;b}c\;=\;\frac ac\;+\;\frac bc $$
Example Of Dividing Polynomials By Monomial
Consider the following example word problem:
Mary Anne bought 6 identical shirts at the store and spent an additional $28 for a new pair of jeans. She had a coupon that gave half of her total purchase. If she ended up paying a total of $48, what was the original cost of each shirt (before the half-off coupon)?
Let x = the original cost of each shirt. Before using the coupon, Mary Anne spent 6x + 28 for the six shirts and pair of jeans. However, since she had a coupon that gave her half off, we must divide the total cost of her purchase by 2, thus we have the equation:
$$ \frac{6x\;+\;28}2\;=\;48 $$
This can be simplified by dividing each term on the left side of the equation by 2 so that we have
$$ \frac{6x\;+\;28}2\;=\;\frac{6x}2\;+\;\frac{28}2 $$
Performing the division, we have,
$$ 3x\;+\;14\;=\;48 $$
To solve for x, we must subtract 14 from each side of the equation:
$$ 3x\;=\;48\;-14 $$
Next, we divide by 3, to obtain
$$ x\;=\;\frac{34}3\;=\;11\frac13 $$
Thus, the original price for each shirt was $11.33.
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. The following property of exponents is useful when a polynomial is divided by a variable raised to some power:
$$ \frac{x^a}{x^b}\;=\;x^{a-b} $$
Example 2:
$$ \frac{20x^3\;-\;80x^2\;+\;16x}{4x} $$
First, we divide each term in the numerator by the term in the denominator,
$$ \frac{20x^3}{4x}\;-\;\frac{80x^2}{4x}\;+\;\frac{16x}{4x} $$
Then using the property of exponents (or simply canceling like terms), we can reduce the expression to,
$$ 5x^2\;-\;20x+\;4 $$
4 Ways To Dividing Polynomials:
1. Long Division:
It is a formal method of division. Also know as bus stop method. We follow 4 steps in long division
- Divide: Take the dividend (Here – 24.60) and the divisor (Here – 12) and do normal division. Here you can take 24 and then divide it by 12.
- Quotient: Write the quotient above the line as shown in the above figure, which is 2 in this case.
- Subtraction: Do the multiplication (divisor x Quotient = 12 x 2 = 24) and then perform subtraction. Here 24 – 24 = 0
- Bring down: write down the remainder and bring the next number down which is 6 here.
Keep repeating the process until you are not able to bring down any number.
2. Synthetic Division:
It is used to manually perform the division of polynomials. This method uses less number of steps compared to long division method. It is also as short hand method.
In synthetic division method, we follow these steps
- Write down the coefficients of the dividend. Here 2x^3 – 5x^2 -x + 3 is the dividend and has descending powers, meaning x has degrees starting from 3 up to 0. Note: 3 can also be written as 3x^0. If the power is missing, then simply add that power with a coefficient as 0. Why add a coefficient of 0? This is because this doesn’t change the meaning of polynomials. coefficient 0 means a number is multiplied by 0 and any number when multiplied by 0 gives 0. So it doesn’t change the dividend in polynomials.
- Use the root associated with the divisor. Divisor is x + 3 so if I considered x + 3 = 0 we get x = -3. Now place -3 and the coefficients of all the dividends as shown in the figure.
- Bring the first number down as it is ( Here 2). and then multiply it with -3, -3 x 2 = -6.
- Place the number below next number and add. Keep repeating the process till you reach last digit of the number to be multiped with.
- Number obtained at the end is the remainder. Keep it aside and starting from right place the variable with powers starting from 0. so 32 becomes 32x^0, -11 becomes -11x, 2 becomes 2x^2. So the quotient is 2x^2-11x+32.
- In case of remainder, it is always the number divided by the actual divisor. -93 here is remainder and x+3 is the number to be divided with. so remainder becomes -93/(x+3)
3. Splitting Division Method:
The splitting division method is another method to divide. In this method, you split the numbers. Say 144/8 can be split into the form 80 + something. You can use any number that is completely divisible by 8. Using 80 is simple as 8 x 10 = 80. so the number becomes 80 + 64. Dividing each number by 8 gives 10 and 8 respectively. As there is an addition between 80 and 64 we need to add 8 and 10 to get the final solution. This method is useful for dividing polynomials with a number. In this case, if I have 144x/8, we need to perform division similar way as we did without worrying about the variable that 144 has. In the end, you can just place the variables back so the solution would be 18x.
This can be applied to a polynomial as well say (144x^2 + 64x) / 8, divide 144/8 separately so answer is 18x^2 and 64x/8 which is 8x so the final answer is 18x^2 + 8x
4.Factorization Method:
As the name suggests breaking polynomial into factors and then dividing it is called factorization method.
In this method write down the polynomial in the form of factors. This can be done by various methods and by applying formulas.
What is the difference between long division and short division?
In long division, we follow the steps divide, multiply, subtract, and bring down.
In short division, you do the multiplication and subtraction steps mentally (or with a calculator, or on scratch paper.
The difference obtained after division is brought down in long division while the difference is written next to the next digit instead of writing the difference below and bringing down the next digit to its level in short division. The calculations involved are the same.
Because more of the work is done mentally in the short division, the long division tends to be preferred over the short division for larger divisors.
