Polynomial Division

Division of Polynomial

A polynomial can be divided by a monomial if the monomial is a factor of the polynomial. The quotient is a polynomial and the remainder is a polynomial.

Concept of Polynomials

A polynomial is an expression consisting of variables and coefficients, that is, numbers that are multiplied together. The coefficients are the numbers that are written in front of the variables. The variables are the letters that represent the unknown quantities.

For example, the polynomial x2 + 5x – 4 can be read as “x squared plus 5x minus 4.” This polynomial has three coefficients (2, 5, and -4) and one variable (x).

Polynomials can be added, subtracted, multiplied, and divided. The operations are performed just as they would be with regular numbers. For example, the polynomial x2 + 5x – 4 can be rewritten as (x + 2)2 – (3x – 4). This polynomial can be further simplified to x2 + 9 – 3x.

Polynomials can be transformed into factored form, which is a form that makes the polynomial easier to work with. The factored form of a polynomial is created by factoring out the greatest common factor of all the terms in the polynomial. For example, the factored form of x2 + 5x – 4 is (x + 2) (x – 2).

Types of Polynomial Division

There are three types of polynomial division: synthetic division, long division, and partial fraction decomposition.

Synthetic division is a method for dividing one polynomial by another using a simple algorithm. The dividend (the polynomial to be divided) is written above the divisor (the polynomial that will do the dividing) with a line drawn between them. The coefficients of the dividend are written below the line, and the coefficients of the divisor are written above the line. The division is then performed by repeatedly subtracting the coefficients of the divisor from the coefficients of the dividend, and writing the result below the line. When the dividend is reduced to zero, the division is complete and the Remainder is the result.

Long division is a method for dividing one polynomial by another using a more complex algorithm. The dividend is written above the divisor with a line drawn between them. The coefficients of the dividend are written to the right of the line, and the coefficients of the divisor are written to the left of the line. The division is then performed by repeatedly subtracting the coefficients of the divisor from the coefficients of the dividend, and writing the result to the right of the line. When the dividend is reduced to zero, the division is complete and the Quotient is the result. The Remainder is the value of the dividend that was not divided evenly by the divisor

Division of One Monomial by Another Monomial

To divide one monomial by another monomial, divide the coefficients of the monomials and keep the terms with the same variables in the same order.

Division of Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

For example, to divide the polynomial 6×3 + 9×2 – 15x + 5 by 3x, divide each term of the polynomial by 3x.

6×3 + 9×2 – 15x + 5

÷ 3x

2×3 + 3×2 – 5x + 1.5

Division of a Polynomial by a Polynomial

If the degree of the polynomial on the left is less than or equal to the degree of the polynomial on the right, then the polynomial on the left can be divided by the polynomial on the right.

The polynomial on the left is divided by the polynomial on the right to get the following:

The quotient is the polynomial on the left divided by the polynomial on the right.

The remainder is the polynomial on the left minus the polynomial on the right.

Exercise –

2

A. True
B. False

A. True