Polynomials – Definition – Example
Polynomials – Definition – Example
A ‘Polynomial’ is a mathematical expression consisting of variables (indeterminates), coefficients and the operations of addition, subtraction, multiplication and non-negative integer exponents of variables.
Example: A Polynomial of a single variable x is
x2 + 3x
A Polynomial of a variable y is
2y3 + 3y +5
Similarly, A Polynomial of a variable m is
m2 – 3m – 5
A Polynomial of a three variables x ,y and z are
x2 – 3xyz + yz + 5
In the polynomial x2 + 3x, the expressions x2 and 3x are called the terms of the polynomial.
Similarly, In the polynomial 2y3 + 3y + 5 the expressions 2y3 , 3y and 5 are called the terms of the polynomial.
The polynomial x2 + 3x, has two terms, x2 and 3x.
The polynomial 2y3 + 3y + 5 has three terms, y3 , 3y, and 5.
The polynomial 2y3 + 3y2 + 5y – 4 has four terms, 2y3 , 3y2, 5y and -4.
Each term of a polynomial has a coefficient, so, in the polynomial
x2 + 3x, the coefficient of x2 is 1 and coefficient of x is 3. in the polynomial 2y3 + 3y + 5 the coefficient of y3 is 2, coefficient of y is 3, and 5 is the coefficient of is x0 because (x0 = 1).
5, -5, -4, 2, 3 etc. are also examples of constant polynomials.
The constant polynomial zero is called the zero polynomial.
If the variable in a polynomial is x, we denote the polynomial by p(x).
If the variable in a polynomial is y, we denote the polynomial by q(y) etc.
p(x) = x2 + 3x + 2
q(y) = y5 – y4 + 2y
Degree of a polynomial
In a polynomial the highest power of the variable is called the ‘Degree of the polynomial.’
Classification of polynomials on the basis of degree.
1. Zero(0) degree polynomials are called ‘Constants polynomial.’
2. 1st degree polynomials are called ‘Linear polynomial.’
3. 2nd degree polynomials are called ‘Quadratic polynomial.’
4. 3rd degree polynomials are called ‘Cubic polynomial.’
5. 4th degree polynomials are called ‘Biquadratic polynomial.’
Degree Polynomial Example
a. 0 Constant 5, -5 etc.
b. 1 Linear x + 2, 2x + 5 etc.
c. 2 Quadratic x2 + 3x + 2, y2 + 6y etc.
d. 3 Cubic x3 + 3x, y3 + 2y etc.
e. 4 Biquadratic x4 + 3x, y4 + 2y etc.
Examples of degree of polynomials:
1. Find the degree of the each of the polynomials given below.
a. x8 – x4 + 3x – 7
The highest power of the variable is 8. so, the degree of the polynomial is 8.
b. t9 – t2 + 5t + 2
The highest power of the variable is 9. so, the degree of the polynomial is 9.
c. y5 – y4 + 2y
The highest power of the variable is 5. so, the degree of the polynomial is 5.
d. y6+ 3y5 – 5y4 + 2y
The highest power of the variable is 6. so, the degree of the polynomial is 6.
e. 7
Here is only one term that is 7. Which can be written as 7x0 , so the exponent of x is 0.
Therefore, the degree of the polynomial is 0.
Classification of polynomials on the basis of number of terms.
Number of terms Polynomial Example
a. 1 Monomial 4, 5x, etc
b. 2 Binomial 2x + 5, x – 2y
c. 3 Trinomial x2 + 3x + 2
Classification of polynomials on the basis of their uses.
1. Zero degree are used to describe quantities that don’t change.
2. 1st degree polynomials are used to describe quantities that change at a steady rate. They are also used in many one-dimensional geometry problems involving length.
3. 2nd degree polynomials are used to describe quantities that change with some amount of acceleration or deceleration. They are also used in many two-dimensional geometry problems involving area.
4. 3rd degree polynomials are used in many three-dimensional geometry problems involving volume.