Polynomial functions are fundamental elements in mathematics, representing expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. These functions may contain multiple algebraic terms including constants, variables of different degrees, coefficients, and positive exponents.
Given below is the example Polynomial Function:
Key Characteristics of Polynomials Functions:
- Degree: The degree of the polynomial is the highest exponent of the variable.
- Coefficients: These are the numbers that multiply each power of x.
- Constant Term: The term that does not involve x, such as +1 in the example, is the constant term.
The generalized form of the polynomial function is defined as:
P(x) = anxn + an-1xn-1 + ...... + a2x2 + a1x + a0
Where,
- an, an-1, . . . a2, a1, a0 are coefficients,
- x is variable, and
- P(x) is the polynomial function in variable x.
The exponents of the variable should be the whole number. an, an-1,...... a2, a1, a0 coefficients are real number constants. n is a positive number which is the degree of the polynomial. an is the leading coefficient as it is the coefficient of the degree of the polynomial. The leading coefficient in the polynomial function cannot be zero.
Polynomials are generally represented by P(x). Polynomial functions have many terms as in polynomial, poly means many, and nominal means terms. The exponent of the polynomial function must be positive. The domain of the polynomial function is all the real numbers R.
Degree of Polynomial Function
The degree of polynomials function is the highest power of the variable in the function.
Examples
- In the polynomial 5x3 + 4x2 + 2x, the term with the highest exponent is 5x3, so the degree is 3.
- In the polynomial 7y5 - 3y2 + 6, the highest exponent is 7y5, so the degree is 5.
- A constant polynomial, like 6, is considered to have a degree of 0 since there is no variable.
Example: Find the Degree of polynomial function P(x) that is given as follows: P(x) = 4x3 + 3x2 + 2x + 1.
- Degree of term 4x3 is 3,
- Degree of term 3x2 is 2,
- Degree of term 2x is 1, and
- Degree of term 1 is 0.
As the highest degree in all the terms is 3.
Thus, the degree of the above polynomial function P(x) is 3.
Types of Polynomial Functions
We can classify the polynomial functions based on various parameters such as the number of terms it contains or their degree. Classification of polynomial functions based on these parameters is given below:
Based on the Number of Terms
Based on the Number of Terms, a polynomial Function can be classified as follows:
Types of Polynomials | Number of Terms | Examples |
|---|
Monomials | Polynomial functions with only one term. | f = 2xy, g = 4a2, h = -3b3c2 |
Binomial | Polynomial functions with only two terms. | f = 3x + 2y, g = x2 + 2xy, h = 4m2 - 6n |
Trinomial | Polynomial functions with only three terms. | f = 2x + 3y - z, g = x2 + 5xy - 2y2, h = 3m3 + 2m2n - mn2 |
Based on the Degree
Some examples of the polynomial functions are:
- Linear Polynomial (Degree 1):
- P(x) = 2x + 3
- This is a first-degree polynomial (linear function) with a slope of 2 and a y-intercept of 3.
- Quadratic Polynomial (Degree 2):
- P(x) = x2 − 4x + 4
- This is a second-degree polynomial (quadratic function) representing a parabola.
- Cubic Polynomial (Degree 3):
- P(x) = x3 − 3x2 + 2x
- This is a third-degree polynomial (cubic function).
- The graph of this function will have one or more inflection points.
- Quartic Polynomial (Degree 4):
- P(x) = 4x4 − x3 + 2x2 − 5x + 1
- This is a fourth-degree polynomial (quartic function).
- The graph of this polynomial can have up to three turning points.
- Quintic Polynomial ( Degree 5):
- P(x) = x5 − 2x4 + x − 1
- This is a fifth-degree polynomial (quintic function).
- The graph may have up to four turning points and complex behaviors.
- Higher-Degree Polynomial ( Degree 6):
- P(x) = 2x6 − 3x4 + x3 − 2x + 5
- This is a sixth-degree polynomial.
- The graph may have more complex behavior, with multiple turning points.
- Constant Polynomial ( Degree 0):
- P(x) = 7
- This is a zero-degree polynomial (constant function).
- The graph is a horizontal line at y = 7.
- Polynomial with Multiple Variables:
- P(x, y) = 3x2y − 2xy2 + xy
- This is a polynomial function involving more than one variable (x and y).
- The degree is determined by the sum of the exponents of the variables in each term.
Learn more about Types of Polynomials
Polynomial Functions Graphs
Polynomial Functions are graphed in many ways depending on the degree of the given polynomial function. Some polynomial functions are graphed as a line, some as parabolas, and some higher-degree polynomial functions are graphed as curves intersecting the x-axis various times.
Let's understand these graphs individually in detail.
Function | Definition |
|---|
Constant Polynomial Function | The constant polynomial function is of the form P(x) = a. In the graph, a horizontal line represents the constant polynomial function. |
Linear Polynomial Function | The linear polynomial function is of the form P(x) = ax + b. In the graph, a straight line (with slope a and intercept b) represents the linear polynomial function. |
Quadratic Polynomial Function | The quadratic polynomial function is of the form P(x) = ax2 + bx + c. In the graph, a parabola represents the quadratic polynomial function. |
Higher Polynomial Function | The higher polynomial function is of the form P(x) = anxn + an-1xn-1 + ...... + a2x2 + a1x + a0. In the graph, a straight line can intersect the graph on n points if the polynomial function is of degree n. |
Read More: Graph of Polynomial Functions
Roots of Polynomial Function
The roots of the polynomial functions are the numbers that satisfy the equation P(x) = 0. Zeros are also called the roots of the polynomial function or the intercepts of the polynomial function. The roots of the polynomial function are also known as Zeroes of the polynomial.
We have to put P(x) = 0 and solve the equation to obtain the required roots of the polynomial function P(x).
Polynomial function | Maximum Number of roots |
|---|
Linear | 1 |
Quadratic | 2 |
Cubic | 3 |
Quartic | 4 |
Learn more about how to calculate the Zeros of Polynomials
How to Identify a Polynomial Function?
To check whether a given function is polynomial or not, there are some Rules, that are given as follows:
- The exponent of the function should be a positive number. It should not be negative or fractional.
- The variable should not be radical.
- The denominator should not contain any variable.
Also, Read
Solved Problems of Polynomial Function
Let's solve some problems on Polynomial functions.
Problem 1: Identify whether the function is a polynomial function or not.
- Q(x) = 5x-9 + 2
- P(x) = 2x1/2 + 3
- F(x) = 4x3 + 7
Solution:
- Q(x) = 5x-9 + 2 is not a polynomial function as it has negative exponent.
- P(x) = 2x1/2 + 3 is not a polynomial function as it has fractional exponent.
- F(x) = 4x3 + 7 is a polynomial function as it has positive exponent.
Problem 2: Find the zeros of the polynomial function, P(x) = 3x - 15.
Solution:
To find the roots for the polynomial function we have to equate it to 0.
P(x) = 0
⇒ 3x - 15 = 0
⇒ 3x = 15
⇒ x = 5
The root of polynomial function P(x) = 3x -15 is 5.
Problem 3: Find the zeros of the polynomial function, P(x) = x2 + 5x + 6.
Solution:
To find the roots for the polynomial function we have to equate it to 0.
P(x) = 0
x2 + 5x + 6 = 0
⇒ x2 + 3x + 2x + 6 = 0
⇒ x(x+3) + 2(x + 3) = 0
⇒ (x + 3) (x + 2) = 0
⇒ x = -2, -3
The roots of polynomial function P(x) = x2 + 5x + 6 are -3 and -2.
Problem 4: Find the zeros of the polynomial function, P(x) = x3 - 5x2 - x + 5.
Solution:
To find the roots for the polynomial function we have to equate it to 0.
P(x) = 0
x3 - 5x2 - x + 5 = 0
⇒ x2(x - 5) -1(x - 5) = 0
⇒ (x2 - 1) (x - 5) = 0
⇒ (x - 1) (x + 1) (x - 5) = 0
⇒ x = 1, -1, 5
The roots of polynomial function P(x) = x3 - 5x2 - x + 5 are 1, -1, 5.
Practice Questions on Polynomials
Question 1: Identify the degree and type of the polynomial: 3x4 − 5x3 + 2x − 7.
Question 2: Determine if the following expression is a polynomial. If it is, state its degree and type: 2x−1 + 4x2 – 3.
Question 3: Find the degree and type of the following polynomial: 6y2 + 8y – 10.
Question 4: Identify whether the following is monomial, binomial, or trinomial, and state its degree: z3 – 7z + 1.
Question 5: Determine the degree and classify the following polynomial: 9a5b2 – 3a4b + 7ab3.
Question 6: Find the zeroes of the polynomial: p(x) = x2 – 5x + 6.
Question 7: Determine the zeroes of the polynomial: q(x) = 2x2 + 7x – 3.
Question 8: If x = 3 is one of the zeroes of the polynomial r(x) = x3 – 4x2 + x + 6, find the other zeroes.
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