We have studied algebraic expressions and polynomials. To recall an algebraic expression f(x) of the form f(x) = a0 + a1x + a2x2 + a3 x3 + ……………+ an xn, there a1, a2, a3…..an are real numbers and all the index of ‘x’ are non-negative integers is called a polynomial in x.Polynomial comes from “poly” meaning “many” and “nomial” meaning “term” combinedly it means “many terms”A polynomial can have constants, variables and exponents.
The degree of a polynomial is nothing but the highest degree of its exponent(variable) with non-zero coefficient. Here the term degree means power. In this article let us study various degrees of polynomials.
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What is the Degree of a Polynomial?
The highest degree exponent term in a polynomial is known as its degree.
To find the degree all that you have to do is find the largest exponent in the given polynomial.
For example, in the following equation:
f(x) = x3 + 2x2 + 4x + 3. The degree of the equation is 3 .i.e. the highest power of the variable in the polynomial is said to be the degree of the polynomial.
f(x) = 7x2 – 3x + 12 is a polynomial of degree 2.
thus,f(x) = an xn + an-1 xn-1 + an-2xn-2 +……………….+ a1 x + a0 where a0 , a1 , a2 …….an are constants and an ≠ 0 .
On the basis of the degree of a polynomial , we have following names for the degree of polynomial.
Degree of Zero Polynomial
If all the coefficients of a polynomial are zero we get a zero degree polynomial. Any non – zero number (constant) is said to be zero degree polynomial if f(x) = a as f(x) = ax0 where a ≠ 0 .The degree of zero polynomial is undefined because f(x) = 0, g(x) = 0x , h(x) = 0x2 etc. are equal to zero polynomial.
A polynomial having its highest degree zero is called a constant polynomial. It has no variables, only constants.
For example: f(x) = 6, g(x) = -22 , h(y) = 5/2 etc are constant polynomials. In general f(x) = c is a constant polynomial.The constant polynomial 0 or f(x) = 0 is called the zero polynomial.
A polynomial having its highest degree one is called a linear polynomial.
For example, f(x) = x- 12, g(x) = 12 x , h(x) = -7x + 8 are linear polynomials.
In general g(x) = ax + b , a ≠ 0 is a linear polynomial.
A polynomial having its highest degree 2 is known as a quadratic polynomial.
For example, f (x) = 2x2 – 3x + 15, g(y) = 3/2 y2 – 4y + 11 are quadratic polynomials.
In general g(x) = ax2 + bx + c, a ≠ 0 is a quadratic polynomial.
A polynomial having its highest degree 3 is known as a Cubic polynomial.
For example, f (x) = 8x3 + 2x2 – 3x + 15, g(y) = y3 – 4y + 11 are cubic polynomials.
In general g(x) = ax3 + bx2 + cx + d, a ≠ 0 is a quadratic polynomial.
A polynomial having its highest degree 4 is known as a Bi-quadratic polynomial.
For example, f (x) = 10x4 + 5x3 + 2x2 – 3x + 15, g(y) = 3y4 + 7y + 9 are quadratic polynomials.
In general g(x) = ax4 + bx2 + cx2 + dx + e, a ≠ 0 is a bi-quadratic polynomial.
Based on the degree of the polynomial the polynomial are names and expressed as follows:
Types of Polynomials Based on their Degrees
How to Find the Degree of a Polynomial?
There are simple steps to find the degree of a polynomial they are as follows:
Example: Consider the polynomial 4x5+ 8x3+ 3x5 + 3x2 + 4 + 2x + 3
Step 1: Combine all the like terms variables
(4x5 + 3x5) + 8x3 + 3x2 + 2x + (4 + 3)
Step 2: Ignore all the coefficients and write only the variables with their powers.
x5 + x3 + x2 + x + x0
Step 3: Arrange the variable in descending order of their powers if their not in proper order.
x5 + x3 + x2 + x1 + x0
Step 4: Check which the largest power of the variable and that is the degree of the polynomial
x5 + x3 + x2 + x + x0 = 5
1. What is the Degree of the Following Polynomial
i) 5x4 + 2x3 +3x + 4
Ans: degree is 4
ii)11x9 + 10x5 + 11
Ans: degree is 9
2. Find the Zeros of the Polynomial.
p(x) = 3x – 2
3x – 2 = 0
3x = 2
x = ⅔ is a zero of p(x) = 3x – 2
1. Write the Degrees of Each of the Following Polynomials.
12-x + 2x3
4x3 + 2x2 + 3x + 7
2. Identify the Polynomial
p(x) = 2x2 – x + 1
h(x) = x4 + 3x3 + 2x2 + 3