Polynomial Division & Long Division Algorithm


Polynomials is an algebraic expression of the type \(a_{n} x^n + a_{n-1} x^{n-1} +…………………a_{2} x^2 + a_{1} x + a_{0}\),

Where \(n\) is either 0 or positive variables and real coefficients. In this expression \(a_{n},a_{n-1}…..a_{1},a_{0} \) are coefficients of the terms of polynomial.

Highest power of x in the above expression is known as degree of the polynomial.

If \(p(x)\) represents a polynomial and \(x\) = \(k\) such that \(p(k)\) = \(0\) then k is the root of the given polynomial.

Let’s Work Out-

Example: Given polynomial equation, \(p(x)= x^{2} – x – 2\), find the zeros of the equation.

Solution: Given Polynomial, \(p(x)= x^{2} – x – 2\)

Zeros of the equation is given by:

\( x^{2} – 2x + x – 2 = 0\)

\( x(x-2) + 1(x – 2)\)

\( (x+1) (x-2) = 0 \)

\(\Rightarrow x = -1 \; or \; x = 2 \)

Thus -1 and 2 are zeros of the given polynomial.

It is to be noted that the highest power(degree) of the polynomial gives the number of zeros of the polynomial.

Division of Polynomial:

Division is the process of splitting a quantity into equal amounts. In terms of mathematics, the process of repeated subtraction or the reverse operation of multiplication is termed as division. For example when 20 is divided by 4 we get 5 as the result since 4 is subtracted 5 times from 20. The four basic operations viz. addition, subtraction, multiplication and division can also be performed on algebraic expressions.

Let us discuss about dividing polynomials and algebraic expressions.

For dividing polynomials, generally three cases can arise:

  • Division of a monomial by another monomial
  • Division of a polynomial by monomial
  • Division of a polynomial by another polynomial

Let us discuss all these cases one by one:

  • Division of a monomial by another monomial

Consider the algebraic expression 40×2 is to be divided by 10x then

\(\frac{40x^{2}}{10x} = \frac{2 \times 2 \times 5 \times 2 \times x \times x}{2\times 5 \times x}\)

Since 2, 5 and x are common in both the numerator and the denominator.

Hence \(\frac{40x^{2}}{10x} = 4x\)

  • Division of a polynomial by monomial

The second case is when a polynomial is to be divided by a monomial. For dividing polynomials, each term of the polynomial is separately divided by the monomial (as described above) and the quotient of each division is added to get the result. Consider the following example:

Example: Divide 24×3 – 12xy + 9x by 3x.

Solution: The expression 24×3 – 12xy + 9x has three terms viz. 24×3, – 12xy and 9x. For dividing polynomials each term is separately divided as:

\(\frac{24x^3 – 12xy + 9x}{3x} = \frac{24x^{3}}{3x} – \frac{12xy}{3x} + \frac{9x}{3x}\)

\(= 8x – 4y + 3\)<

  • Division of a polynomial by polynomial

For dividing a polynomial with another polynomial, the polynomial is written in standard form i.e. the terms of the dividend and the divisor are arranged in decreasing order of their degrees. Let us take an example.

Example: Divide 3×3 – 8x + 12 by x- 1.

Solution: The Dividend is 3×3 – 8x + 12 and the divisor is x -1.

After this the leading term of the dividend is divided by the leading term of the divisor i.e. 3×3 ÷ x =3×2. This result is multiplied by the divisor i.e. 3×2(x -1) = 3×3 -3×2 and it is subtracted from the divisor. Now again, this result is treated as dividend and same steps are repeated until the remainder becomes zero or its degree becomes less than that of the divisor as shown below.

Dividing Polynomials

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