Question

Which of the following is quadratic polynomial
A) \[x + 2\]
B) \[{x^2} + 2\]
C) \[{x^3} + 2\]
D) \[2x + 2\]

Answer 1

Hint:
Here we will use the concept of a quadratic polynomial to answer this question. We will first define the quadratic equation. Then we will analyze all the equations one by one and find which of the equations is correct.

Complete step by step solution:
To answer this question, we should know what a quadratic polynomial actually is.
A quadratic polynomial is a polynomial with power 2 and it can be written in the form of \[a{x^2} + bx + c\] having two solutions.
Now, we will analyze the options one by one.
A) \[x + 2\]
This is clearly not a quadratic polynomial because it cannot be written in the form of \[a{x^2} + bx + c\] as the power of variable \[x\]is 1.
This is however, a linear polynomial with power 1 as it can be written in the form of \[ax + b\].

B) \[{x^2} + 2\]
As we can see that the power of variable \[x\] is 2 and this polynomial can easily be written in the form of \[a{x^2} + bx + c\] where \[a = 1\], \[b = 0\] and \[c = 2\]
Hence, clearly this is a quadratic polynomial.

C) \[{x^3} + 2\]
As we can see that the power of variable \[x\] is 3 and this polynomial cannot be written in the form of \[a{x^2} + bx + c\]hence, it is not a quadratic polynomial. In fact, this is a cubic polynomial which can be written in the form of \[a{x^3} + b{x^2} + cx + d\] with degree 3.

D) \[2x + 2\]
As part A. this polynomial is also a linear polynomial as the degree of variable \[x\]is 1. And also, this can be written in the form of \[ax + b\] instead of \[a{x^2} + bx + c\]. Hence, it is not a quadratic polynomial.

Therefore, option B is the required answer as \[{x^2} + 2\] is a quadratic polynomial.

Note:
We should know the basic differences between different types of polynomials like a linear polynomial has a degree 1, a quadratic has a degree 2, a cubic has a degree 3 and so on. Similarly, a linear has only 1 solution, quadratic has 2 solutions, cubic has 3 solutions and so on. These differences are very important for solving various questions in mathematics in less time. Also, knowing their general equations help us to find the roots easily by using the relationship between the coefficients and the roots.