Question

Given 5 examples of linear polynomial, quadratic and cubic polynomial.

Answer 1

Hint: Each of the polynomials has a specific degree and based on that they have been assigned a specific name and are thus referred to as different types of polynomials. There are four types of polynomials – constant, linear, quadratic, and cubic polynomials. We have to give examples of each type of polynomial.

Complete step-by-step answer:
We know that,
Polynomials are classified on degree and here, the term degree means power. And this further determines the maximum number of solutions a function could have and the number of times a function will cross the x-axis when graphed.
A polynomial having degree zero is called Constant polynomial. In general \[f\left( x \right) = c\] is a constant polynomial.

The types of polynomial are below:
1) Linear Polynomials:-
A linear polynomial is a polynomial of degree one, i.e., the highest exponent of the variable is one.
Here, we define an equation in the form:
\[p\left( x \right):ax + b,{\text{ }}a \ne 0\]
Given below are a few examples of linear polynomials:
I.\[x + 3\]
II.\[4x + \dfrac{2}{5}\]
III.\[ - 2x\]
OV.\[x + \sqrt 2 \]
V.\[\pi x + \sqrt 3 \]
We note that a linear polynomial in one variable can have at most two terms if ‘a’ is 0, then this will become a constant polynomial.

2) Quadratic Polynomials:-
A quadratic polynomial is a polynomial of degree two, i.e., the highest exponent of the variable is two. In general, a quadratic polynomial will be of the form:
\[p\left( x \right):a{x^2} + bx + c,{\text{ }}a \ne 0\]
Given below are a few examples of quadratic polynomials:
I.\[2{x^2} + 2x + 1\]
II.\[{x^2} - 4\]
III.\[\sqrt 2 {x^2}\]
IV.\[4{x^2} + \dfrac{1}{7}\]
V.\[2-{x^2} + x\surd 3\]
VI.\[{x^2}\surd 5{\text{ }} + {\text{ }}\dfrac{2}{3}x{\text{ }}-6\]
We observe that a quadratic polynomial can have at most three terms and if ‘a’ is 0, then this will become a linear polynomial.

3) Cubic Polynomials:-
A cubic polynomial is a polynomial of degree three, i.e., the highest exponent of the variable is three. A cubic polynomial, in general, will be of the form:
\[p\left( x \right):a{x^3} + b{x^2} + cx + d,{\text{ }}a \ne 0\]
Given below are a few examples of cubic polynomials:
I.\[2{\text{ }}-{\text{ }}{x^3}\]
II.\[{\text{81}}{x^3}\; - {\text{ 5}}\]
III.\[\pi {x^3} + {\left( {\sqrt 2 } \right)^{11}}\]
IV.\[{x^3}\]
V.\[3{x^3}\;-{\text{ }}2{x^2}\; + {\text{ }}x{\text{ }}-{\text{ }}1\]
VI.\[{x^3}\surd 2\]
We observe that a cubic polynomial can have at most four terms and if ‘a’ is 0, then this will become a quadratic rather than a cubic polynomial.

Note: After converting any expression into the general form, if the exponent of the variable in any term is not a whole number, then it's not a polynomial either. A polynomial of degree n will have n number of zeros or roots. A linear polynomial has only one zero. A quadratic polynomial can have at most two zeros, whereas a cubic polynomial can have at most 3 zeroes.