Physics and Mathematics
Polynomial Function
1. Concept Overview
A polynomial function is one of the most important and most commonly used functions in mathematics.
Almost all basic ideas of:
- limits,
- continuity,
- differentiation, and
- graph sketching
are first understood using polynomial functions.
2. Definition of Polynomial Function
A function [f] is called a polynomial function if it is of the form:
[f(x)=aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀]
where:
- [aₙ, aₙ₋₁, … , a₀] are real constants
- [n] is a non-negative integer
- [aₙ ≠ 0]
3. Degree of a Polynomial Function
The highest power of x in the polynomial is called the degree.
Examples:
| Polynomial | Degree |
|---|---|
| [f(x)=7] | 0 |
| [f(x)=3x+1] | 1 |
| [f(x)=x²−4x+5] | 2 |
| [f(x)=2x³−x+1] | 3 |
4. Types of Polynomial Functions
(a) Constant Polynomial Function
[f(x)=c]
- Degree = 0
- Graph is horizontal
(b) Linear Polynomial Function
[f(x)=ax+b]
- Degree = 1
- Graph is a straight line
(c) Quadratic Polynomial Function
[f(x)=ax²+bx+c]
- Degree = 2
- Graph is a parabola
(d) Cubic Polynomial Function
[f(x)=ax³+bx²+cx+d]
- Degree = 3
- Graph is S-shaped
5. Domain and Range
Domain:
Every polynomial function is defined for all real numbers.
So,
Domain = [R]
Range:
- Depends on the degree and nature of the polynomial
- Even degree → range may be restricted
- Odd degree → range is usually all real numbers
6. Important Properties of Polynomial Functions
- Always continuous
- Always differentiable
- No breaks or gaps in graph
- Can be one–one or many–one
- Graphs are smooth curves
7. Polynomial Function and Roots
If [f(a)=0], then:
- [a] is called a root or zero of the polynomial
- Graph cuts or touches the x-axis at [(a,0)]
8. Conceptual Questions with Solutions
1. What is a polynomial function?
A polynomial function is a function expressed as a finite sum of powers of x with real coefficients.
2. Why must powers of x be non-negative integers?
Because negative or fractional powers would make the function non-polynomial.
3. Is [f(x)=1/x] a polynomial function?
No. It has a negative power of [x], so it is not a polynomial.
4. Is [f(x)=\sqrt{x}] a polynomial?
No. It involves a fractional power of [x].
5. Why are polynomial functions always continuous?
Because they are formed using basic algebraic operations that preserve continuity.
6. Can a polynomial function be many–one?
Yes. Some polynomial functions give the same output for different inputs.
7. Is every polynomial function differentiable?
Yes. Polynomial functions are differentiable everywhere.
8. What determines the shape of a polynomial graph?
The degree and leading coefficient determine the shape.
9. What is a zero of a polynomial?
A zero is a value of [x] where the function becomes zero.
10. Can a polynomial have no real roots?
Yes. Some polynomials have no real zeros.
11. Is [f(x)=0] a polynomial?
Yes. It is a zero polynomial.
12. Can degree of polynomial be negative?
No. Degree is always a non-negative integer.
13. Why is polynomial important in calculus?
Because it is easy to differentiate and analyze.
14. Is identity function a polynomial?
Yes. It is a polynomial of degree one.
15. Can domain of polynomial be restricted?
Yes. Domain can be restricted artificially if specified.
9. FAQ / Common Misconceptions
1. Every algebraic function is polynomial.
False. Some algebraic functions are not polynomial.
2. Polynomial functions can have gaps.
False. They are continuous everywhere.
3. Polynomial functions are not differentiable.
False. They are differentiable everywhere.
4. Degree zero polynomial is not useful.
False. Constant polynomials are very useful.
5. Polynomial functions always have roots.
False. Some have no real roots.
6. Fractional powers are allowed.
False. They make the function non-polynomial.
7. Polynomial graphs are zig-zag.
False. They are smooth curves.
8. Degree decides number of roots exactly.
False. It gives maximum possible roots.
9. Leading coefficient has no role.
False. It controls the end behavior.
10. Polynomial functions are only theoretical.
False. They have practical applications.
10. Practice Questions
Question 1.
Find the degree of [f(x)=3x⁴−5x²+7].
Step-by-Step Solution:
- Highest power of [x] is [4].
Conclusion:
Degree = 4
Question 2.
State whether [f(x)=x³−x] is one–one or many–one.
Step-by-Step Solution:
- [f(1)=f(−1)=0].
- Different inputs give same output.
Conclusion:
The function is many–one.
Question 3.
Find the domain of [f(x)=x²+4x+1].
Step-by-Step Solution:
- Polynomial functions are defined for all real [x].
Conclusion:
Domain = [R]
Question 4.
Find [f′(x)] for [f(x)=2x³−x+5].
Step-by-Step Solution:
- Differentiate term by term.
Conclusion:
[f′(x)=6x²−1]
Question 5.
Find [f(2)] if [f(x)=x²−3x+4].
Step-by-Step Solution:
- Substitute [x=2].
Conclusion:
[f(2)=2]
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