Question

Describe third degree polynomials.

Answer 1

Hint: First, we should be familiar with the concept of polynomials in order to answer this question in a better way. The term polynomial basically consists of two words i.e. ${\text{poly + nomial}}$ . Poly means ‘many’ and the word nominal means ‘terms’. Therefore the word polynomial means ‘many terms’. Some examples of polynomials are: $3{x^4} + {x^3} + 2{x^2} + 5x + 7$ , $9{x^3} + 5{x^2} + 2$ , $6x$ , $5{x^2} + 3$. Polynomials consist of variables and their coefficients and the power of variables are always non-negative integers.

Complete step by step answer:
Let us first see the types of polynomials, they can be classified in two ways;
$\left( 1 \right)$ On the basis of number of terms present in a polynomial, it can be categorized in three types;
$\left( a \right)$ Monomial: Only one term is present. Example: $6x,{\text{ }}2,{\text{ }}2{a^4}$ etc.
$\left( b \right)$ Binomial: Two terms are present. Example: $ - 9x + 2,{\text{ 6}}{{\text{x}}^2} + 11x{\text{ }},{\text{ }}x + 1$ etc.
$\left( c \right)$ Trinomial: Three terms are present. Example: $4{x^3} + 3x + 2,{\text{ }}8{x^2} + 9x + 5,{\text{ }}10{x^2} + 2x + 3$ etc.

$\left( 2 \right)$ On the basis of degree of a polynomial, it can be categorized into;
Before that, let us first understand the concept of degree of a polynomial.
Degree of a polynomial: The degree of a polynomial can be defined as the highest power of a term or monomial within a polynomial. In other words , the greatest degree of a variable in any polynomial is referred to as the degree of the polynomial. For example: $9{x^4} + 4{x^3} + 1$ is a polynomial , so let us find out it’s degree. The first term is $9{x^4}$, the power of the variable $x$ is $4$. The second term is $4{x^3}$, here the power of the variable is $3$. The last term is a constant term and it does not have any variable, therefore the power of the variable is $0$. We have to choose the highest power, therefore the degree of this polynomial will be $4$ . Now let us move further and see the classification;
$\left( a \right)$ Constant or Zero polynomial: A constant term is present. Example: $4$, no variable is present ; therefore degree is zero. So it is basically a polynomial with degree $0$ .
$\left( b \right)$Linear polynomial: A polynomial whose degree is $1$ , is called a linear polynomial. Example:
$2x + 1$ .
$\left( c \right)$ Quadratic polynomial: A polynomial with degree equals to $2$ , is called a quadratic polynomial. Example: $5{x^2} + 2x + 1$ .
$\left( d \right)$ Cubic polynomial : A polynomial whose degree is $3$, is known as a cubic polynomial or a degree $3$ polynomial. Example: $4{x^3} + 7{x^2} + x + 1$ .

As asked in the question, let us discuss the cubic polynomial or degree $3$ polynomial in detail.
There is a very important point about the degree of a polynomial. If the degree of a polynomial is $x$ ,
Then the polynomial will have $x$ roots or zeros i.e. Number of roots = Degree of a polynomial
Like in case of cubic polynomials the degree is $3$ , therefore the number of roots for the cubic
polynomial will also be $3$ . Let the roots or zeros of a cubic polynomial are $\alpha ,{\text{ }}\beta ,{\text{ }}\gamma $ . There exists a
relationship between the coefficient of a cubic polynomial and its roots or zeros. A cubic polynomial is
generally represented by; $a{x^3} + b{x^2} + cx + d$, where $a \ne 0$ , otherwise it will become a quadratic
polynomial. The relationship between zeroes and coefficient of a cubic polynomial are given by;
$\left( a \right){\text{Sum of roots = }}\alpha {\text{ + }}\beta {\text{ + }}\gamma {\text{ = }} - \dfrac{b}{a}$
$\left( b \right){\text{Sum of product of roots}}\left( {{\text{two at time}}} \right){\text{ = }}\alpha \beta {\text{ + }}\beta \gamma + \alpha \gamma {\text{ = }}\dfrac{c}{a}$
$\left( c \right){\text{Product of roots = }}\alpha \beta \gamma {\text{ = }} - \dfrac{d}{a}$

Note: There are also some rules which should be satisfied by an expression to be a polynomial. An
expression should not consists of: $\left( 1 \right)$ The square root of variable or variable inside the
radical sign , example: $\sqrt x $ . $\left( 2 \right)$ There can not be negative power on variables,
example: $7{x^{ - 4}}$ . $\left( 3 \right)$ There can not be fractional power present on variables,
example: ${x^{\dfrac{3}{2}}}$ . $\left( 4 \right)$ Variables can not be in the denominator of any
fractions, example: $\dfrac{1}{{{x^3}}}$ . If an expression fails to satisfy these rules then it is not a polynomial.