Dividing Polynomials
Polynomials are algebraic expressions that comprise coefficients, and variables. It is represented in the form of 7x² - 6x + 17. This polynomial has three terms that are arranged as per their degrees. The term with the highest degree in the polynomial expression is placed first, followed by the lower ones. In algebra, the division of polynomials is an algorithm to divide a polynomial by another of the same or lower degree.
Dividing polynomials (either with a long division or synthetic division) is similar to long division in Arithmetic. We divide the dividend by the divisor to get a quotient and the remainder (which will be equal to 0 if the divisor is the factor for dividend).
In other words, given a dividend f(x), divisor g(x), quotient p(x), and remainder q(x). We know: \[\frac{f(x)}{g(x)} = p(x) + \frac{q(x)}{g(x)}\]. With polynomial division long, the division process continues until the remainder q(x) is reached whose degree is less than the degree of the divisor g(x).
Long Division Method of Polynomial Steps
Following are the steps of the long division method of polynomials:
The first step is to arrange the given terms in the descending order of their indices (if required). Also, write the missing term with 0 as their coefficients.
The second step is to divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
The third step is to multiply the divisor by the quotient to obtain the remainder value. Also, subtract the product from the original dividend to get the remainder. and bring down the next term if any.
In the fourth step, the difference and the term that is brought down will form a new dividend.
Now, repeat the process till you get a remainder, which can be zero or of a lower index than a divisor.
Let's now understand the long division method polynomials steps through an example for better understanding.
Long Division Method Polynomial Example
We need to divide the polynomial a(x) = 12 - 14x² - 13x by (2x + 3)
First, we will arrange the terms in the descending order of the power of their variables.
a(x): -14x² -13 x - 12
b(x): (2x + 3)
Divide a(x) by b(x)in a similar way as we perform the regular division.
\[\frac{-14x^{2} - 13x -12}{(2x + 3)}\]
Add the missing indices with 0 as their coefficients.
Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient as shown below:
\[\frac{-14x^{2}}{2x}\]
We get, -7x
Multiplying the divisor by the quotient Also, subtracting the product from the original dividend to get the remainder.
2x + 3 ) -14x2 - 13x + 2( -7x
-14x2 - 21x ↓
(+) (+)
-----------------------------
8x + 12 ← (bring down the next term)
↙
(new dividend)
The remainder obtained will be treated as a new dividend, leaving the divisor the same.
Now, we will divide the first term of the new dividend by the first term of the divisor to get the second term of the quotient. Also, we will multiply the divisor by the second term of the quotient to get the product and subtract the product that is just obtained from the new dividend to get the remainder as shown below.
2x + 3 ) -14x2 - 13x + 2( -7x
-14x2 - 21x ↓
(+) (+)
-----------------------------
8x + 12
8x + 12
(-) (-)
-----------------------------
0
Therefore,
\[\frac{-14x^{2} - 13x - 12}{(2x + 3)} = -7x + 4\]
Also, it is also concluded that divisor and quotient are the factors of the dividend as the remainder is 0.
Division Algorithm For Polynomials
The division algorithm for polynomials states that if p(x), and q(x)are two polynomials, where q(x) ≠ 0, we can write the division algorithm for polynomials as
p(x) = k(x) × q(x) + r(x)
Where,
p(x) is the dividend.
k(x) is the divisor.
q(x) is the quotient.
r(x) is the remainder.
This is similar to the regular division of numbers, where we follow the rule Dividend= (Divisor × Quotient ) + Remainder to verify the answer of division.
Division Algorithm For Polynomials Prove
Using the long division method of dividing polynomial by another polynomial, let us divide 3x³ + x² + 2x + 5 by x² + 2x + 1.
With this division of polynomial, we can now verify the division algorithm for polynomials as:
p (x)= 3x³ +x² + 2x + 5
q(x)= x² + 2x + 1
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Also, quotient g(x)= 3x - 5
Remainder r(x) = 9 x + 10
Division algorithm for polynomials states that p(x) = g(x) × q(x)+ r(x)
Substituting the values of dividend, divisor, remainder and quotient in the above equation, we get”
p(x) = g(x) × q(x) + r(x)
3x³ +x + 2x + 5 = (x² + 2x + 1) × (3x - 5) + 9 x - 10
3x³ +x + 2x + 5 = (3x³ +6x² + 3x - 5x² -10 x - 5) + (9 x + 10)
3x³ +x + 2x + 5 = 3x³ +6x² + 3x -5x² -10 x - 5 + 9 x + 10
3x³ +x + 2x + 5 = 3x³ +6x² -5x² + 3x -10x+ 9x - 5+ 10
3x³ +x + 2x + 5 = 3x³ +x² + 2x+ 5
Hence, p (x) = g(x) × q(x)+ r(x)
Therefore, the division algorithm for polynomials is verified.
Dividing Polynomials by Monomials
A monomial is an algebraic expression that consists of only one non-zero term. For example, x is a monomial in one variable ‘x’.
Dividing Polynomials by monomials means dividing the polynomial ( which is considered as a numerator value) by monomials (which is considered as the divisor value) to find the quotient value.
Example:
4x³ - 10x² + 5x 2x
Here, the polynomial 4x³ - 10x² + 5x is considered as a numerator and the monomials 2x are considered as the denominator.
Hence, we get
\[\frac{4x^{3} - 10x^{2} + 5x}{2x}\]
Here, we can see there are three terms in a polynomial, so each term of the polynomial (numerator) is separately divided by the same monomial (denominator).
\[\frac{4x^{3}}{2x} - \frac{4x^{2}}{2x} + \frac{5x}{2x}\]
Now, we will cancel out the common factor from both numerator and denominator to simplify the equation.
Therefore, we get
\[= 2x^{2} - 5x + \frac{5}{2}\]
Dividing Polynomials by Binomials
Binomial is an algebraic expression that consists of only two terms. For example, (x + 3) is a binomial expression.
Dividing polynomials by binomial with long division can be done easily by following the below steps:
Arrange the terms of both the dividend and divisor in the descending order of their exponents.
Further, divide the first term of the dividend by the first term of the divisor in order to get the first term of the quotient.
Multiply the divisor by the first term of the quotient and subtract the result from the dividend to get the remainder.
Now, consider this remainder as a new dividend and repeat step (2) to get the second term of the quotient.
Repeat the process till we get the reminder value equal to zero or a polynomial of degree less than the divisor.
Dividing Polynomials by Binomials Example
Let us understand dividing polynomials by binomials with an example.
Divide (x² + 7x + 12) by (x + 4)
Step 1: Look at the first term of both the dividend and divisor and divide the terms as follows:
\[\frac{x^{2}}{x} = x\].
We will write the quotient (x) at the top to carry forward the long division and multiply (x) by ( x + 4) to get the second term of our solution as shown below.
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Step 2: Subtracting the second row from the first row gives,
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Step 3: Bringing down the remaining term i.e. 12 as shown below:
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Step 4: As the remainder is 3x + 12. So, multiply (x + 4) by 3 and consider 3 as a quotient value. The 3 will be written at the top as shown below.
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Step 5: Subtract: (3x + 12) and (3x + 12) as shown below
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Therefore, (x² + 7x + 12) ÷ (x + 4) = x + 3
You can verify your answer by multiplying (x + 3) by ( x + 4) , you will get (x² + 7x + 12).
Solved Examples
1. Divide x³ + 9x - 6x² - 2 by (x - 2)
Solution:
Rearrange the terms of both the dividend and divisor in descending order and divide it as follows:
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Hence, the quotient is x² - 4x + 1, and the remainder is 0.
2. Divide x² + xy - xz by - x.
Solution:
= x² + xy - xz ÷ - x
Dividing each term of the polynomial by the monomial and simplifying it.
\[= \frac{x^{2}}{-x} + \frac{xy}{-x} - \frac{xz}{-x}\]
\[= \frac{-x^{2}}{x} - \frac{xy}{x} + \frac{xz}{x}\]
Now, each term will be simplified by canceling out the common factors. Therefore, we get:
= -x - y + z
Therefore, x² + xy - xz - x = -x - y + z
3. Divide 6a⁴ + 5a³ + 4a - 4 by 2a² + a - 1
Solution:
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As (4a - 3) has a lower degree than 2a² + a - 1, we can stop the division process here. Hence, we can conclude that 3a² + a + 1 is the quotient polynomial and 4a - 3 is the remainder polynomial. This implies that
6a⁴ + 5a³ + 4a - 4 = (2a² + a - 1) ( 3a² + a + 1) + 4a - 3.
FAQs on Polynomials - Long Division
Q1. What is Synthetic Division?
Ans. Synthetic division is the shortcut way to determine the quotient and remainder values when dividing polynomials with a monic linear binomial (a polynomial in the form of (x - k). The most important aspects of synthetic division are that it enables one to calculate the quotient and remainder values without writing variables and, with fewer calculations.
Q2. How do You Come to Know if the Divisor is the Factor of Polynomials?
Ans. If the value of the remainder r(x) equals 0, then we can say that there is no remainder. This implies that D(x) = d(x)q(x). The equation states that divisor is the multiple of dividend or the divisor can divide the dividend completely. With this, we can say that that divisor and also the quotient is a factor of the dividend.
Q3. What are the Rules for Dividing Polynomials by Monomials?
Ans. When dividing polynomials by the monomial expressions, consider the monomial expression as a common denominator and split the fraction using the following property.
(x + y)/z = x/z + y/z
Using this property results in terms that can be considered as quotient.
