Division Algorithm for Polynomials: Definition, Diagram, Types, Examples

In this article, we shall learn about the Division Algorithm for Polynomials, along with many examples. A polynomial is an algebraic expression with a term or terms consisting of real number coefficients and the variable factors with the whole numbers exponents. The degree of a polynomial is the highest value of the variable’s exponent among its terms (sum of the variables if the terms contain more than one variable).

Division is a mathematical operation in which things are divided into equal portions. It’s also known as the multiplication operation’s inverse. Polynomial division involves multiplying one polynomial by a monomial, binomial, trinomial, or lower degree polynomial. The dividend has a higher degree than the divisor in a polynomial division. We multiply the divisor and the quotient and add them to the remainder to verify the result. Continue reading to know more.

What is a Polynomial?

Definition: Let \(x\) be a variable, \(n\) be a positive integer and \(a_{1}, a_{2}, \ldots, a_{n}\) are constants (real numbers). Then, \(f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}\) is known as a polynomial in variable \(x\).

The terms of a polynomial are connected by the mathematical operations – addition \((+)\) or subtraction \((-)\).

Examples: \(7 x+3,11 y-6\) are examples of Polynomials.

Division Algorithm for Polynomials

There are two methods by which a polynomial can be divided by another polynomial:

a) Long Division Method
b) Division involving factorising either or both dividend and divisor using algebraic identity 

Long Division Method

Definition: When we divide \(3 x^{2}\) by \(x\), we get
\(\frac{3 x^{2}}{x}=3 x\), here Dividend \(=3 x^{2}\), Divisor \(=x\), Quotient \(=3x\) and Remainder \(=0\)

So, \(3 x^{2}=x \times 3 x+0\)

Here, the division algorithm for the polynomials can be written as shown below:

\({\text{Dividend}} = ({\text{Divisor}} \times {\text{Quotient}}) + {\text{Remainder}}\)

Usually, if \(p(x)\) and \(g(x)\) are the two polynomials and that degree of \(p(x) \geq\) degree of \(g(x)\) and \(g(x) \neq 0\), then you can find the polynomials \(q(x)\) and \(r(x)\) such that:

\(p(x)=g(x) q(x)+r(x)\)

Here, \(r(x)=0\) or the degree of \(r(x)<\) degree of \(g(x)\). We can say that \(p(x)\) divided by \(g(x)\), gives \(q(x)\) as the quotient and \(r(x)\) as the remainder.

Example: Find the quotient of \(\left(x^{3}-8 x^{2}+19 x-12\right) \div(x-1)\).

First, you will write the dividend which you are dividing, \(x^{3}-8 x^{2}+19 x-12\), inside the long division bar and outside, you have to write the divisor, dividing by \(x-1\).

You will start with the most significant power terms \(x\) and \(x^{3}\).

Here, write the term with which you will multiply \(x\) to generate \(x^{3}: x^{2}\)

Now, multiply the divisor by the term \(\left(x^{2}\right)\) and place the result below the dividend, so the like terms are aligned.

Later subtract the new polynomial from the real dividend. Write down the rest of the words to acquire the latest dividend.

Here, write the next term which you will multiply \(x\) to generate \(-7 x^{2}:-7x\).

Again, write the next term, which you will multiply \(x\) to generate \(12 x: 12\).

After multiplying both the terms in the divisor by the number \(12\), arrange the like terms and then subtract where everything is cancelled and get no remainder, i.e., zero.

Hence, in this case, we obtain quotient \(=x^{2}-7 x+12\) and remainder \(=0\).

In general, if \(f(x)\) and \(g(x)\) are two polynomials with \(g(x) \neq 0\), then we can find polynomials \(q(x)\) and \(r(x)\) such that:

\(f(x)=g(x) \cdot q(x)+r(x)\), where \(r(x)=0\) or \(\operatorname{deg} r(x)<\operatorname{degg}(x)\)

The above process is known as the Division Algorithm of Polynomials.

If \(r(x)=0\), then \(g(x)\) is called a factor of \(f(x)\).

If \(\operatorname{deg} f(x) \geq \operatorname{degg}(x)\), then \(\operatorname{deg} q(x)=\operatorname{deg} f(x)-\operatorname{degg}(x)\).

Division Using Factorisation Using Algebraic Identity

In this method, we use standard algebraic identities to factorise either or both the polynomials given as dividend and divisor and cancel the common factors from both numerator and the denominator.

Example: We can divide \(x^{3}-8\) by \(x-2\) as follows:

\(\frac{x^{3}-8}{x-2}=\frac{(x)^{3}-(2)^{3}}{x-2}=\frac{(x-2)\left[(x)^{2}+x \times 2+(2)^{2}\right]}{x-2}=\frac{(x-2)\left[x^{2}+2 x+4\right.}{(x-2)}\)
\(=x^{2}+2 x+4\)

Here, the algebraic identity \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\) is used.

Formulas Used for Division of Algebraic Expressions

We give below some of the standard algebraic identities that are used for factorisation to divide algebraic expressions.

1. \(a^{2}-b^{2}=(a-b)(a+b)\)
2. \((a+b)^{2}=a^{2}+2 a b+b^{2}\)
3. \((a-b)^{2}=a^{2}-2 a b+b^{2}\)
4. \((a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a\)
5. \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=a^{3}+b^{3}+3 a b(a+b)\)
6. \((a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}=a^{3}-b^{3}-3 a b(a-b)\)
7. \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\)
8. \(a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\)
9. \(a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)\)

Working Rule to Divide a Polynomial by Another Polynomial

a) If \(\operatorname{deg} f(x)<\operatorname{deg} g(x)\) then \(f(x)=g(x) \cdot 0+f(x)\), i.e., the quotient is \(0\) and \(r(x)=f(x)\).

b) If \(\operatorname{deg} f(x) \geq \operatorname{deg} g(x)\), then arrange the terms of both polynomials in standard form. Divide the first term of \(f(x)\) by the first term of \(g(x)\) to get the first term of \(q(x)\).
Multiply the whole divisor, i.e., \(g(x)\), by the first term of the quotient and then subtract it from the dividend to get the remainder.

c) If the \(\operatorname{degr}(x) \geq \operatorname{degg}(x)\), treat this remainder as the new dividend and proceed as above.
Repeat till \(r(x)=0\) or \(\operatorname{degr}(x)<\operatorname{degg}(x)\).

d) If the final remainder is \(r(x)\) and the quotient is \(q(x)\), then \(f(x)=g(x) \cdot q(x)+r(x)\), when either \(r(x)=0\) or \(\operatorname{degr}(x)<\operatorname{deg} g(x)\).

Solved Examples– Division Algorithm for Polynomials

Q.1. Divide the polynomial \(f(x)=14 x^{3}-5 x^{2}+9 x-1\) by the polynomial \(g(x)=2 x-1\). Also, find the quotient and remainder and verify the division algorithm.
Ans: Using the long division method, we obtain

Quotient \(q(x)=7 x^{2}+x+5\) and remainder \(r(x)=4\).
Now, \(q(x) g(x)+r(x)=\left(7 x^{2}+x+5\right)(2 x-1)+4\)
\(=14 x^{3}+2 x^{2}+10 x-7 x^{2}-x-5+4\)
\(=14 x^{3}-5 x^{2}+9 x-1\)
\(=f(x)\)
i.e \(f(x)=g(x) q(x)+r(x)\) or \({\text{Dividend}} = {\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}}\).

Q.2. Divide the polynomial \(f(x)=6 x^{3}+11 x^{2}-39 x-65\) by the polynomial \(g(x)=x^{2}-1+x\). Also, find the quotient and remainder and verify the division algorithm.
Ans: Using the long division method, we obtain

Quotient \(q(x)=6 x-17\) and remainder \(r(x)=62 x-82\)
Also, \(g(x) q(x)+r(x)=\left(x^{2}+x-1\right)(6 x-17)+(62 x-82)\)
\(=6 x^{3}+6 x^{2}-6 x-17 x^{2}-17 x+17+62 x-82\)
i.e., \(f(x)=g(x) q(x)+r(x)=6 x^{3}-11 x^{2}+39 x-65=f(x)\)
or, \({\text{Dividend}} = {\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}}\).

Q.3. Divide the polynomial \(f(x)=3 x^{2}-x^{3}-3 x+5\) by the polynomial \(g(x)=x-1-x^{2}\) and verify the division algorithm.
Ans: Writing the given polynomial in standard form, we get
\(f(x)=-x^{3}+3 x^{2}-3 x+5\) and \(g(x)=-x^{2}+x-1\)
Using the long division method, we obtain

\(\therefore\) Quotient \(q(x)=x-2\) and, remainder \(r(x)=3\)
Now,
\({\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}} = (x – 2)\left( { – {x^2} + x – 1} \right) + 3\)
\(=-x^{3}+x^{2}-x+2 x^{2}-2 x+2+3\)
\(=-x^{3}+3 x^{2}-3 x+5=\) dividend
Hence, the division algorithm is verified.

Q.4. Divide: \(x^{4 a}+x^{2 a} y^{2 b}+y^{4 b}\) by \(x^{2 a}+x^{a} y^{b}+y^{2 b}\)
Ans: We have,
\(x^{4 a}+x^{2 a} y^{2 b}+y^{4 b}\)
\(=\left(x^{2 a}\right)^{2}+x^{2 a} y^{2 b}+\left(y^{2 b}\right)^{2}\)
\(=\left(x^{2 a}\right)^{2}+2 x^{2 a} y^{2 b}+\left(y^{2 b}\right)^{2}-x^{2 a} y^{2 b}\)
\(=\left(x^{2 a}+y^{2 b}\right)^{2}-\left(x^{a} y^{b}\right)^{2}\)
\(=\left(x^{2 a}+y^{2 b}+x^{a} y^{b}\right)\left(x^{2 a}+y^{2 b}-x^{a} y^{b}\right)\)
\(=\left(x^{2 a}+x^{a} y^{b}+y^{2 b}\right)\left(x^{2 a}-x^{a} y^{b}+y^{2 b}\right)\)
\(\therefore \frac{x^{4 a}+x^{2 a} y^{2 b}+y^{4 b}}{x^{2 a+x^{a}} y^{b}+y^{2 b}}=\frac{\left(x^{2 a}+x^{a} y^{b}+y^{2 b}\right)\left(x^{2 a}-x^{a} y^{b}+y^{2 b}\right)}{x^{2 a+x^{a} y^{b}+y^{2 b}}}=x^{2 a}-x^{a} y^{b}+y^{2 b}\)

Q.5. Divide the polynomial \(u(x)=9 x^{4}-4 x^{2}+4\) by the polynomial \(v(x)=3 x^{2}+x-1\). Also, find the quotient and remainder and verify the division algorithm.
Ans: Using the long division method, we obtain

Quotient \(q(x)=3 x^{2}-x\) and remainder \(r(x)=-x+4\).
Now, \(v(x) q(x)+r(x)=\left(3 x^{2}+x-1\right)\left(3 x^{2}-x\right)+(-x+4)\)
\(=9 x^{4}+3 x^{3}-3 x^{2}-3 x^{3}-x^{2}+x-x+4\)
i.e \(\quad u(x)=v(x) q(x)+r(x)=9 x^{4}+0 x^{3}-4 x^{2}+0 x+4=u(x)\)
or, \({\text{Dividend}} = {\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}}\).

Summary

This article covered polynomials with an example, and then we discussed the division algorithm. We explained both the division of polynomials: long division method and division of polynomials by factorisation of either dividend or divisor or both using standard identities with examples. Then, we have also provided few solved examples, followed by FAQs.

Frequently Asked Questions– Division Algorithm for Polynomials

Q.1. Explain division algorithm for polynomials with example.
Ans: When we divide \(6 x^{3}\) by \(2 x^{2}\), we get \(\frac{6 x^{3}}{2 x^{2}}=3 x\), here Dividend \(=6 x^{3}\), Divisor \(=2 x^{2}\), Quotient \(=3 x\) and Remainder \(=0\)
So, \(6 x^{3}=3 x \times 2 x^{2}+0\)
Then, the division algorithm for polynomials can be written as
\({\text{Dividend}} = {\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}}\)
In general, if \(p(x)\) and \(g(x)\) are the two polynomials in case that degree of \(p(x) \geq\) degree of \(g(x)\) and \(g(x) \neq 0\), then we can find polynomials \(q(x)\) and \(r(x)\) such that:
\(p(x)=g(x) q(x)+r(x)\)
Where \(r(x)=0\) or the degree of \(r(x)<\) degree of \(g(x)\). Here we say that \(p(x)\) divided by \(g(x)\), gives \(q(x)\) as quotient and as the remainder it gives \(r(x)\).

Q.2. What is the formula for the division algorithm?
Ans: The formula for the division algorithm can be written as given below:
\({\text{Dividend}} = {\text{Quotient}} \times {\text{Divisor}} + {\text{Remainder}}\)

Q.3. What is a division algorithm? Explain with an example?
Ans: The algorithm is a series of well-defined steps that solve the type of problem.
Division Algorithm: If \(f(x)\) and \(g(x)\) are two polynomials with \(g(x)\) and \(g(x) \neq 0\), then we can find polynomials \(q(x)\) and \(r(x)\) such that:
\(f(x)=g(x) \cdot q(x)+r(x)\) Where \(r(x)=0\) or \(\operatorname{deg} \operatorname{deg} r(x)<\operatorname{degg}(x)\)

Example: We have, \(f(x)=x^{4}-3 x^{2}+4 x+5\) and \(g(x)=x^{2}-x+1\)
We find that degree \((f(x))=4\) and degree \((g(x))=2\). Therefore, quotient \(q(x)\) is of degree \(2(=4-2)\), and remainder \(r(x)\) is of degree less than \(2(=\operatorname{degree}(g(x))\). So, let \(q(x)=a x^{2}+b x+c\) and \(r(x)=p x+q\)
Using division algorithm, we have
\(f(x)=g(x) \times q(x)+r(x)\)
\(\Rightarrow x^{4}+0 x^{3}-3 x^{2}+4 x+5=\left(x^{2}-x+1\right)\left(a x^{2}+b x+c\right)+p x+q\)
\(\Rightarrow x^{4}+0 x^{3}-3 x^{2}+4 x+=a x^{4}+(b-a) x^{3}+(c-b+a) x^{2}+(b-c+p) x+c+q\)
On equating the coefficients of various powers of x on both sides, we get
\(a=1\) [On equating the coefficients of \({x^2}\)]

\(b-a=0\) [On equating the coefficients of \({x^3}\)]

\(c-b+a=-3\) [On equating the coefficients of \({x^2}\)]

\(b-c+p=4\) [On equating the coefficients of \(x\)]

And \(c+q=5\) [On equating the constant terms]

Solving these equations, we get
\(a=1, b=1, c=-3, p=0\) and \(q=8\)
\(\therefore\) Quotient \(q\left( x \right) = {x^2} + x – 3\) and Remainder \(r(x)=8\)

Q.4. How do you find the division of a polynomial?
Ans: Below is the given steps to divide a polynomial:
1. The polynomial should be written in descending order. If you see any terms are missing, then you can use zero instead.
2. Now, divide the term with the highest power inside the division sign with the term with the highest power outside the division sign.
3. So, now you have to multiply the answer you got in the earlier step by the polynomial in front of the division symbol.
4. Then, subtract and get down the next term.
5. You have to repeat the second, third, and fourth steps until you get no more terms to bring down.
6. Finally, write the answer, and the term you get after the subtraction is known as the remainder, and it will be written as the fraction in the final solution.

Q.5. What is the division algorithm for Class \(5\)?
Ans: Division algorithm for class \(5\) is, \({\text{Dividend}} = {\text{Divisor}} \times {\text{Quotient}} \times {\text{Remainder}}\)
Remember, the remainder should always be smaller than the divisor.

Q.6. What is Euclid’s division algorithm formula?
Ans: Euclid’s Division Algorithm applies Euclid’s division lemma several times to obtain the HCF of any two numbers.
Apply Euclid’s division lemma to \(a\) and \(b\) and obtain \(\mid\) whole numbers \(q_{1}\) and \(r_{1}\) such that \(a=b q_{1}+r_{1}, 0<r_{1}<b\)
If \(r_{1}=0, b\) is the \(\mathrm{HCF}\) of \(a\) and \(b\).
If \(r_{1} \neq 0\), apply Euclid’s division lemma to \(b\) and obtain two whole numbers \(q_{1}\) and \(r_{2}\) such that \(b=q_{1} r_{1}+r_{2}\).
\({\text{Dividend}} = ({\text{Divisor}} \times {\text{Quotient}}) + {\text{Remainder}}\)

Q.7. Which operation is used in the division algorithm?
Ans: Subtraction and shift operations are the two primary operations to apply the division algorithm. After each subtraction, the divisor (multiplied by the numbers one or zero) is transferred to the right side by one bit relative to the dividend.