Polynomial and its Types | Algebra - Mathematics PDF Download

Introduction to Polynomial Equation

If p(x) is a polynomial equation in x, then the highest power of x in p(x) is called the degree of the polynomial p(x). So, p(x) =

4x + 2 is a polynomial equation in the variable x of degree 1

2y² – 3y + 4 is a polynomial in the variable y of degree 2

5x³ – 4x² + x – 2 is a polynomial in the variable x of degree 3

7u6 – 3u4 + 4u2 – 6 is a polynomial in the variable u of degree 6

Further, it is important to note that the following expressions are NOT polynomials:

• 1 / (x – 1)
• √x + 2
• 1 / (x² + 2x + 3)

Types of Polynomials

Let’s look at the different types of polynomials that you will come across while studying them.

Linear Polynomials

Any polynomial with a variable of degree one is a Linear Polynomial. Some examples of the linear polynomial equation are as follows:

• 2x – 3
• y + √2
• x √3 + 5
• x + 5/11
• 2/3y – 5

Any polynomial where the degree of the variable is greater than one is not a linear polynomial.

Quadratic Polynomials

Any polynomial with a variable of degree two is a Quadratic Polynomial. The name ‘quadratic’ is derived from the word ‘quadrate’ which means square. Some examples of the quadratic polynomial equation are as follows:

2x² + 3x – 5

y² – 1

2 – x² + x√3

u/3 – 2u² + 5

v2√5 + 2/3v – 6

4z² + 1/7

To generalize, most quadratic polynomials in x are expressed as, ax² + bx + c … where a, b, and c are real numbers. Also, a ≠ 0.

Cubic Polynomials

Any polynomial with a variable of degree three is a Cubic Polynomial. Some examples of the cubic polynomial equation are as follows:

x³

2 – x³

x³√2

x³ – x² + 3

3x³ – 2x² + x – 1

To generalize, most quadratic polynomials in x are expressed as, ax³ + bx² + cx + d … where a, b, c, and d are real numbers. Also, a ≠ 0.

Some more concepts

To begin with, let’s look at the following polynomial p(x),

p(x) = x² – 3x – 4

Next, let’s put x = 2 in p(x). So, we get p(2) = (2)² – 3(2) – 4 = 4 – 6 – 4 = –6. Note that the value ‘– 6’ is obtained by replacing x with 2 in the polynomial x2 – 3x – 4. Hence, it is called ‘the value of x² – 3x – 4 at x = 2’. Similarly, p(0) is the value of x² – 3x – 4 at x = 0.

Therefore, we can say, If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k and is denoted by p(k).
Zero of a Polynomial

So, what is the value of p(x) = x² – 3x – 4 at x = – 1? p(– 1) = (– 1)2 – {3(– 1)} – 4 = 1 + 3 – 4 = 0. Also, the value of p(x) = x² – 3x – 4 at x = 4 is, p(4) = (4)2 – 3(4) – 4 = 16 – 12 – 4 = 0. In this case, since p(– 1) and p(4) is equal to zero, ‘– 1’ and ‘4’ are called zeroes of the quadratic polynomial x² – 3x – 4.

Therefore, we can say, A real number k is said to be a zero of a polynomial p(x) if p(k) = 0. In the previous years, you have already studied how to find zeroes of a polynomial equation. To elaborate on it a little more, if ‘k’ is a zero of p(x) = 2x + 3, then

p(k) = 0

Or 2k + 3 = 0

i.e. k = – 3/2

Let’s generalize this. If ‘k’ is a zero of p(x) = ax + b, then p(k) = ak + b = 0. Or, k = – b/a. In other words, the zero of the linear polynomial (ax + b) is: – (Constant Term) / (Coefficient of x)

Hence, we can conclude that the zero of a linear polynomial equation is related to its coefficients. You will study more about if this rule is applicable to all types of polynomials discussed above.

Solved Examples for You

Question: What are the three types of polynomials and how are they differentiated?

Solution: The three types of polynomials are:

1. Linear
2. Quadratic
3. Cubic

The linear polynomials have a variable of degree one, quadratic polynomials have a variable with degree two and cubic polynomials have a variable with degree three.