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Polynomials

Polynomial and Its Types

By now you are aware of the polynomial equation in one variable and their degrees. In this article, we will look at the various types of polynomials to establish a foundation for further studies into them.

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Introduction to Polynomial Equation

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.

Browse more Topics under Polynomials

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 1 is not a linear polynomial.

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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 where 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.

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