Elementary Algebra
Polynomial
An expression of the form a0x n + a1x n −1 + a2x n − 2 +...+ an −1x + an ,
where a0 , a1 ,... , an are real numbers and n is a non-negative integer, is
called a polynomial in the variable x. Polynomial in the variable x are
usually denoted by f ( x ), g ( x ) and h( x ) etc.
Thus, f ( x ) = a0x n + a1x n −1 + a2x n − 2 + ... + an −1x + an .
(i) If a0 ≠ 0, then n is called the degree of the polynomial f ( x ); it is
written as deg f ( x ) = n.
(ii) a0x n , a1x n −1 , a2x n − 2 ,... , an −1x, an are called the terms of the
polynomial f ( x ); an is called the constant term.
(iii) a0 , a1 , a2 ,... , an −1 , an are called the coefficients of the polynomial
f ( x ).
(iv) If a0 ≠ 0, then a0x n is called the leading term and a0 is called
the leading coefficient of the polynomial.
(v) If all the coefficients a0 , a1 , a2 ,... , an −1 , an are zero, then f ( x ) is
called a zero polynomial. It is denoted by the symbol 0. The
degree of the zero polynomial is never defined.
Degree of a Polynomial
(i) In One Variable The highest power of the variable is called
the degree of the polynomial.
(ii) In Two Variables The sum of the powers of the variables in
each term is obtained and the highest sum, so obtained is the
degree of that polynomial.
Types of Polynomials
(i) Constant Polynomial A polynomial having degree zero.
(ii) Linear Polynomial A polynomial having degree one.
(iii) Quadratic Polynomial A polynomial having degree two.
(iv) Cubic Polynomial A polynomial having degree three.
(v) Biquadratic Polynomial A polynomial having degree four.
, Fundamental Operations on Polynomial
(i) Addition of Polynomials To calculate the addition of two
or more polynomials, we collect different groups of like powers
together and add the coefficients of like terms.
(ii) Subtraction of Polynomials To find the subtraction of two
or more polynomials, we collect different groups of like powers
together and subtract the coefficient of like terms.
(iii) Multiplication of Polynomials Two polynomials can be
multiplied by applying distributive law and simplifying the like
terms.
(iv) Division of Polynomials When a polynomial p ( x ) is
divided by a polynomial q( x ) ≠ 0, we get two polynomials g( x ) and
r( x ) such that
p ( x ) = q( x )g( x ) + r( x )
Synthetic Division Method (Horner’s Method)
This method is to find the quotient and the remainder when a
polynomial is divided by a binomial.
Rule for Synthetic Division
1. First complete the given polynomial f ( x ) by adding the missing
term with zero coefficients.
2. Write the successive coefficients a0 , a1 , a2 ,... , an of the
polynomial f ( x ).
3. If we want to divide the polynomial by x − h, then write h in the
left corner.
4. In third row write b0 below a0, where b0 = a0 and then multiply b0
by h to get the product hb0.
5. Adding hb0 to a1, we get b1. Similarly by adding hb1 to a2, we get
b2 and so on
h a0 a1 a2 ……… an
+ hb0 + hb1 ………
b0 b1 b2 ………
6. Repeat this till you get last term which is remainder R.
If R = 0, then h is the root of the polynomial f ( x ) = 0 and the
equation can be reduced by one dimension.
Polynomial
An expression of the form a0x n + a1x n −1 + a2x n − 2 +...+ an −1x + an ,
where a0 , a1 ,... , an are real numbers and n is a non-negative integer, is
called a polynomial in the variable x. Polynomial in the variable x are
usually denoted by f ( x ), g ( x ) and h( x ) etc.
Thus, f ( x ) = a0x n + a1x n −1 + a2x n − 2 + ... + an −1x + an .
(i) If a0 ≠ 0, then n is called the degree of the polynomial f ( x ); it is
written as deg f ( x ) = n.
(ii) a0x n , a1x n −1 , a2x n − 2 ,... , an −1x, an are called the terms of the
polynomial f ( x ); an is called the constant term.
(iii) a0 , a1 , a2 ,... , an −1 , an are called the coefficients of the polynomial
f ( x ).
(iv) If a0 ≠ 0, then a0x n is called the leading term and a0 is called
the leading coefficient of the polynomial.
(v) If all the coefficients a0 , a1 , a2 ,... , an −1 , an are zero, then f ( x ) is
called a zero polynomial. It is denoted by the symbol 0. The
degree of the zero polynomial is never defined.
Degree of a Polynomial
(i) In One Variable The highest power of the variable is called
the degree of the polynomial.
(ii) In Two Variables The sum of the powers of the variables in
each term is obtained and the highest sum, so obtained is the
degree of that polynomial.
Types of Polynomials
(i) Constant Polynomial A polynomial having degree zero.
(ii) Linear Polynomial A polynomial having degree one.
(iii) Quadratic Polynomial A polynomial having degree two.
(iv) Cubic Polynomial A polynomial having degree three.
(v) Biquadratic Polynomial A polynomial having degree four.
, Fundamental Operations on Polynomial
(i) Addition of Polynomials To calculate the addition of two
or more polynomials, we collect different groups of like powers
together and add the coefficients of like terms.
(ii) Subtraction of Polynomials To find the subtraction of two
or more polynomials, we collect different groups of like powers
together and subtract the coefficient of like terms.
(iii) Multiplication of Polynomials Two polynomials can be
multiplied by applying distributive law and simplifying the like
terms.
(iv) Division of Polynomials When a polynomial p ( x ) is
divided by a polynomial q( x ) ≠ 0, we get two polynomials g( x ) and
r( x ) such that
p ( x ) = q( x )g( x ) + r( x )
Synthetic Division Method (Horner’s Method)
This method is to find the quotient and the remainder when a
polynomial is divided by a binomial.
Rule for Synthetic Division
1. First complete the given polynomial f ( x ) by adding the missing
term with zero coefficients.
2. Write the successive coefficients a0 , a1 , a2 ,... , an of the
polynomial f ( x ).
3. If we want to divide the polynomial by x − h, then write h in the
left corner.
4. In third row write b0 below a0, where b0 = a0 and then multiply b0
by h to get the product hb0.
5. Adding hb0 to a1, we get b1. Similarly by adding hb1 to a2, we get
b2 and so on
h a0 a1 a2 ……… an
+ hb0 + hb1 ………
b0 b1 b2 ………
6. Repeat this till you get last term which is remainder R.
If R = 0, then h is the root of the polynomial f ( x ) = 0 and the
equation can be reduced by one dimension.
