Question

 Identify constant, linear, quadratic and cubic polynomials from the following polynomials:
(i) \[f(x)=0\]
(ii) \[g(x)=2{{x}^{3}}-7x+4\]
(iii) \[h(x)=-3x+\dfrac{1}{2}\]
(iv) \[p(x)=2{{x}^{2}}-x+4\]
(v) \[q(x)=4x+3\]
(vi) \[r(x)=3{{x}^{3}}=4{{x}^{2}}+5x-7\]

Answer 1

Hint: We will have to know about constant, linear, quadratic and cubic polynomials first. If the maximum exponent of a variable is 0, then it is a constant polynomial. If the maximum exponent is 1, then it is a linear polynomial. If the maximum exponent of a variable is 2, then it is a quadratic polynomial and for a cubic polynomial, the maximum exponent is 3.

Complete step-by-step answer:
 We will check all the polynomials, one by one as follows:
(i) \[f(x)=0\]
Since the maximum exponent of a variable in the above polynomial is zero, hence it is a constant polynomial.
(ii) \[g(x)=2{{x}^{3}}-7x+4\]
Since the maximum exponent of a variable is 3, hence it is a cubic polynomial.
(iii) \[h(x)=-3x+\dfrac{1}{2}\]
Since the maximum exponent of a variable is 1, hence it is a linear polynomial.
(iv) \[p(x)=2{{x}^{2}}-x+4\]
Since the maximum exponent of a variable is 2, hence it is a quadratic polynomial.
(v) \[q(x)=4x+3\]
Since the maximum exponent of a variable is 1, hence it is a linear polynomial.
(vi) \[r(x)=3{{x}^{3}}=4{{x}^{2}}+5x-7\]
On taking all the terms to the left hand side of the equal sign, we get as follows:
\[r(x)=3{{x}^{3}}-4{{x}^{2}}-5x+7=0\]
Since the maximum exponent of a variable is 3, hence it is a cubic polynomial.
Therefore, the given polynomials are identified as constant, linear, quadratic and cubic polynomials.

Note: Be careful while observing the maximum exponent of the variable of polynomials. Also remember that a constant polynomial has a maximum exponent of variable as zero, for linear polynomial it is one, for quadratic polynomial it is two and for cubic polynomial it is three.