Polynomial functions

Polynomial Functions

A polynomial function is a mathematical expression that involves variables raised to whole number powers. The general form of a polynomial function in one variable is:

f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

In this expression, a_n, a_{n-1}, ..., a_1, a_0 are constants called coefficients, and n is a non-negative integer that represents the degree of the polynomial. The degree of the polynomial is the highest power of x in the expression.

Types of Polynomial Functions

Polynomial functions can be classified based on their degree:

• Constant Polynomial: A polynomial of degree 0, e.g., f(x) = 5.
• Linear Polynomial: A polynomial of degree 1, e.g., f(x) = 2x + 3.
• Quadratic Polynomial: A polynomial of degree 2, e.g., f(x) = x^2 - 4x + 4.
• Cubic Polynomial: A polynomial of degree 3, e.g., f(x) = x^3 + 2x^2 - x + 1.
• Higher-Degree Polynomials: Polynomials of degree greater than 3, e.g., f(x) = 2x^4 - 3x^3 + x - 7.

Graphing Polynomial Functions

The graph of a polynomial function is smooth and continuous. The shape of the graph depends on the degree of the polynomial and the sign of the leading coefficient (the coefficient of the term with the highest degree).

Example: Graphing a Quadratic Function

Consider the quadratic function:

f(x) = x^2 - 4x + 3

To graph this function, we first find its x-intercepts by solving:

x^2 - 4x + 3 = 0

We can factor this equation:

(x - 1)(x - 3) = 0

The solutions are:

x = 1 and x = 3

Next, we find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

x = -b/(2a)

In our case, a = 1 and b = -4, so:

x = -(-4)/(2 × 1) = 2

Now, substitute x = 2 back into the function to find the y-coordinate of the vertex:

f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1

The vertex is at the point (2, -1). The graph opens upwards since the leading coefficient (1) is positive.

End Behaviour of Polynomial Functions

The end behaviour of a polynomial function describes how the function behaves as x approaches positive or negative infinity. The end behaviour is determined by the degree of the polynomial and the leading coefficient.

• If the degree is even and the leading coefficient is positive, the graph rises to the right and rises to the left.
• If the degree is even and the leading coefficient is negative, the graph falls to the right and falls to the left.
• If the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
• If the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.

Zeros of Polynomial Functions

The zeros (or roots) of a polynomial function are the values of x for which f(x) = 0. These values can be found using various methods, including:

• Factoring
• Using the quadratic formula for quadratic polynomials
• Long division or synthetic division for polynomials of degree greater than 2

Example: Finding Zeros of a Cubic Polynomial

Consider the cubic polynomial:

f(x) = x^3 - 6x^2 + 11x - 6

To find the zeros, we can use the Rational Root Theorem, which suggests that if there are any rational roots, they must be factors of the constant term (-6) divided by factors of the leading coefficient (1). The possible rational roots are:

±1, ±2, ±3, ±6

We can test these values to find the zeros. Testing x = 1:

f(1) = (1)^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0

Since x = 1 is a zero, we can factor f(x) as:

f(x) = (x - 1)(x^2 - 5x + 6)

Now we can factor the quadratic:

f(x) = (x - 1)(x - 2)(x - 3)

The zeros are x = 1, 2, 3.

Summary

• A polynomial function is expressed as f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0.
• The degree of a polynomial determines its shape and end behaviour.
• Zeros of polynomial functions can be found using various methods, including factoring and the Rational Root Theorem.

Check your understanding

1. What is the general form of a polynomial function?
2. How do you determine the end behaviour of a polynomial function?
3. Find the zeros of the polynomial f(x) = x^2 - 5x + 6.
4. Describe the shape of the graph of a cubic polynomial with a negative leading coefficient.