Dividing polynomials in math is an arithmetic operation in which one polynomial is divided by another polynomial, where the divisor polynomial must have a degree less than or equal to the dividend polynomial; otherwise, division of polynomials can't take place.
The most general form of a polynomial is given as:
anxn + an−1xn−1 + ... + a2x2 + a1x + a0
Where a0, a1, a2, ..., and an are the real coefficients. In dividing polynomials, we divide the polynomial with a higher degree by a polynomial (that can be a monomial, binomial, trinomial, or any other higher-degree polynomial) with a lower degree.
There are various methods of dividing polynomials; some of those methods are:
Long Division of Polynomials
The long division method is the most frequent and general method for dividing polynomials by binomials or any other form of polynomials. In case the given numerator and denominator do not have any common factors, you can simplify the expression by using the long division method.
Steps
To divide the polynomial using long division, we can use the following steps:
Step 1: Arrange both Divisors and Dividends in the decreasing order of degree of each of term i.e., anxn + an−1xn−1 + . . . + a1x + a0.
Step 2: Arrange the Divisor and Dividend Long Division Form.
Step 3: Divide the the dividend's first term by the divisor's first term, and use it as the quotient's first term.
Step 4: Multiply the divisor by the result of step 3 and arrange them under the diviant such that like terms aligned with each other.
Step 5: Subtract the result of step 3 from the dividend to create new polynomial.
Step 6: Repeat the steps 3, 4, 5 until the resulting polynomial has a degree less than divisor.
Example 1: Divide x2 + 2x + 3 by x - 2.
Solution:
Step 1: Arrange both Divisors and Dividends in the decreasing order of degree
Dividend = x2 + 2x + 3 Divisor = x - 2
Step 2: We will write the dividend and divisor in the long division form, like this:
Step 3: Divide the first term of the dividend by the first term of the divisor, to get the first term of divisor i.e.,
x2/x = x
This means that the first term of the quotient is x.
Step 4: Multiply the divisor (x - 2) by the first term of the quotient (x), and arrange them under the dividend with like terms aligned.
Step 5: Subtracted the result of step 3 from the dividend to get a new polynomial,
Step 6: We will repeat steps 3 and 4 using the new polynomial 4x + 3:
4x/x = 4
Multiplying divisor (x - 2) by the 4,
Step 7: Since the degree of the new polynomial 11, has a degree 0 which is less than the degree of the divisor (x - 2) i.e., 1, we can stop here.
So, the result of dividing the polynomial x2 + 2x + 3 by x - 2 is:
Quotient: x + 4 and Remainder: 11.
Example 2: Divide x3 + 2x2 - 5x + 1 by x - 2.
Solution:
Synthetic Division of Polynomials
It is a technique for dividing a polynomial by a linear binomial using only the coefficient values. We write the polynomials in standard form from the greatest degree term to the lowest degree term in this manner. Use zero as the coefficients of the missing terms when writing in descending powers.
Steps
To divide the polynomials using synthetic division, we can use the following steps:
Step 1: Write the polynomial in standard form, with the terms arranged in descending order of degree.
Let's Suppose a polynomial as a3x3 + a2x2 + a1x + a0.
Step 2: Identify the divisor, and ensure that the divisor is in the form of x − k
Step 3: Write the value k to the left of the division symbol and place a vertical line to the right of it, and write the coefficients of the polynomial to be divided in the spaces to the right of the vertical line. Where any missing coefficients should be represented by zeros.
Step 4: Bring down the first coefficient and write it below the line, and multiply it with k and write the result in the next space to the right.
Step 5: Add both values in the second space and write it in space next to the leading coefficient.
Step 6: Multiply the result of Step 5 with k and write it in the third space, and add both column values to get the further result.
Step 7: Repeat this till reach the real constant part.
If a0 + k(a1 + k(a2 + ka3)) = 0, then x − k is a factor of f(x). Otherwise, the remainder of the division is a0 + k(a1 + k(a2 + ka3)).
Step 8: The resulting values represent the coefficient of the remainder (last value) and qotient (other values).
Example 1: Divide 3x4 + 5x + 9 by x – 1 using synthetic division.
Write the polynomial in standard form, with the terms arranged in descending order of degree: 3x4 + 0x3 + 0x2 + 5x + 9, Identify the divisor, and ensure that the divisor is in the form of x − k: Divisor = x – 1,
Then follow these steps:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
The resulting values represent the coefficient of the remainder (last value) and quotient (other values).
Therefore, the quotient is 3x3 + 3x2+ 3x + 8 and the remainder is 17.
Example 2: Divide the polynomial x2 + x - 2 by x - 1 by synthetic division.
The first two numbers of the last row represent the coefficients of the quotient and the third value is the remainder.
Thus, the quotient is x + 2 and remainder is 0.
Dividing Polynomial by Monomial
When a polynomial is divided by a monomial, i.e., a polynomial with only one term in it, then the resulting polynomial can only be found if the degree of the divisor is less than or equal to the degree of the polynomial under consideration. We can divide a polynomial by a monomial using the following methods:
1. Using Splitting the Term Method
This method involves splitting each term of the polynomial into separate terms and then simplifying them by dividing each term with the monomial. Let's consider an example to understand this method better.
Example: Divide the polynomial 6x3 + 12x2 + 9x by the monomial 3x.
Therefore, 2x2 + 6x + 4 divided by 2x is equal to x + 3 + 2/x.
Example 3: Divide the polynomial 9x3 - 15x2 + 6x by the monomial 3x.
Solution:
9x3 - 15x2 + 6x = 3x(3x2 - 5x + 2)
Therefore, 9x3 - 15x2 + 6x divided by 3x is equal to 3x2 - 5x + 2.
Dividing Polynomial by Binomial using Factorization
Factorization is the method of writing the given polynomial as a product of its factors. If the binomial by which we are dividing a given polynomial is the factor of the given polynomial, then we can eliminate the binomial, and we are left with the other factor as the quotient. This can be better understood by the example given below:
Example: Divide x2 + 4x + 3 by x + 1
Solution:
We have the Polynomial x2 + 4x + 3 as dividend
The binomial x + 1 is our divisior
We will factorize the polynomial x2 + 4x + 3 using Middle Term Splitting method
Common Mistakes to Avoid When Dividing Polynomials
There are some common mistakes done by students when dividing one polynomial by another. Some of these common mistakes are as follows:
Making an error when performing the division: Double-check your arithmetic when performing the division. It is easy to make a mistake when working with long polynomials, so take your time and be thorough.
Misaligning the terms when setting up the division: Make sure to align the terms correctly when setting up the division. Each term should be in the correct position relative to the other terms in the dividend and divisor.
Forgetting to include the remainder: When dividing polynomials, there may be a remainder. Make sure to include the remainder in your answer, if there is one, and if the remainder is 0, then write 0 in the place of the remainder.
Not simplifying the answer: Finally, simplify the answer as much as possible. This means combining like terms and putting the polynomial in standard form.
