Polynomials

A polynomial is a mathematical expression of the form P(x) = anxn + an-1xn-1+ ...+ a1x + a0 where an, an-1,... are constants and an ≠ 0.

Polynomial means an expression with many terms


Standard form of a Polynomial

When we arrange the polynomial in the decreasing order of its degree, it is said to be in the standard form.  In general, an nth degree polynomial P(x) = anxn + an-1xn-1 + ...+ a1x + a0 where an, an-1,... are constants and an ≠ 0 is in the standard form.  Here we can see that the power of x in each term decreases.
Example: -
P(x)= 4x6 + 3x5 + 9x4 - 3x + 11

Degree of the polynomial

The highest power of the variable in the polynomial is the degree of the polynomial.
Example: -
Let P(x) = 7x8 + 12x5 + 11x2 - 2x + 13
This polynomial is in the standard form.  So the degree is the highest power of the variable in the left most term.  Therefore degree of the polynomial = 8
The polynomial of degree one is called a linear polynomial.  Polynomial of second degree is called Quadratic Polynomial.  And polynomial of degree 3 is known as Cubic polynomial.Lets consider a few examples of converting polynomials to standard form.

Example 1: -
Convert the polynomial to standard form and find its degree.
P(x) = 17x12 - 3x15 + 12x7 - 4x - 13
Solution: -
Given polynomial is P(x) = 17x12 - 3x15 + 12x7 - 4x - 13.
To convert this into standard form, we have to arrange this in the decreasing order of the powers of the variables.
Rearranging the terms, we get
P(x) = - 3x15 + 17x12 + 12x7 - 4x - 13.
This is the required standard form.
Since the polynomial is in the standard form, the degree is the highest power of the variable in the left most term.  Therefore degree of the polynomial = 15

Example 2: -
Convert the polynomial to standard form and find its degree.
P(x) = 13x10 - 23x18 + 72x6 - 4x3 - 19
Solution: -
Given polynomial is P(x) = 13x10 - 23x18 + 72x6 - 4x3 - 19
To convert this into standard form, we have to arrange this in the decreasing order of the powers of the variables.
Rearranging the terms, we get
P(x) = - 23x18 + 13x10 + 72x6 - 4x3 - 19
This is the required standard form.
Since the polynomial is in the standard form, the degree is the highest power of the variable in the left most term.  Therefore degree of the polynomial = 18

Example 3: -
Convert the polynomial to standard form and find its degree.
P(x) = 217x16 + 24x17 - 15x27 - 6x + 16
Solution: -
Given polynomial is P(x) = 217x16 + 24x17 - 15x27 - 6x + 16
To convert this into standard form, we have to arrange this in the decreasing order of the powers of the variables.
Rearranging the terms, we get
P(x) = - 15x27 + 24x17 + 217x16 - 6x + 16
This is the required standard form.
Since the polynomial is in the standard form, the degree is the highest power of the variable in the left most term.  Therefore degree of the polynomial = 27

Try yourself: -
Convert the following polynomials into standard form and find the degree of the following polynomials
a. P(x) = 3x16 + 22x17 + x3 - 16x + 96
b.P(x) = -115x29 + 214x19 + 218x26 - 6x + 36
c.P(x) = - 167x37 - 24x18 + 267x17 + 27x - 46

Answers: -
          Standard form                                              Degree
a. P(x) = 22x17 + 3x16 + x3 - 16x + 96                       17
b. P(x) = -115x29  + 218x26 + 214x19 - 6x + 36           29
c. P(x) = - 167x37 - 24x18 + 267x17 + 27x - 46            37