M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let 1 2 1 0, ,..., , ,−n na a a a a , be real numbers, with 0≠na .

The function defined by 2

2 1 0( ) ,...,= + + + +nnf x a x a x a x ais called a polynomial function of x

of degree n. The number na , the coefficient of the variable to the highest power, is called the

leading coefficient. Note: The variable is only raised to positive integer powers–no negative or fractional exponents. However, the coefficients may be any real numbers, including fractions or irrational numbers

like π or 7 .

Graph Properties of Polynomial Functions Let P be any nth degree polynomial function with real coefficients. The graph of P has the following properties. 1. P is continuous for all real numbers, so there are no breaks, holes, jumps in the graph. 2. The graph of P is a smooth curve with rounded corners and no sharp corners. 3. The graph of P has at most n x-intercepts. 4. The graph of P has at most n – 1 turning points.

3 2( ) ( 2) ( 1)= − +f x x x x

M 1310 4.1 Polynomial Functions 2

Polynomial Not a polynomial Example 1: Given the following polynomial functions, state the leading term, the degree of the polynomial and the leading coefficient. a. 4 3 2( ) 7 5 7 6= − + − +P x x x x x b. 2 3( ) (3 2)( 7) ( 2)= + − +P x x x x End Behavior of a Polynomial

Odd-degree polynomials look like 3= ±y x .

3=y x 3= −y x

M 1310 4.1 Polynomial Functions 3 Even-degree polynomials look like 2= ±y x .

2=y x 2= −y x Power functions: A power function is a polynomial that takes the form ( )= nf x ax , where n is a positive integer.

Modifications of power functions can be graphed using transformations. Even-degree power functions: Odd-degree power functions: 4( ) =f x x 5( ) =f x x

Note: Multiplying any function by a will multiply all the y-values by a. The general shape will stay the same. Exactly the same as it was in section 3.4.

M 1310 4.1 Polynomial Functions 4 Zeros of a Polynomial If f is a polynomial and c is a real number for which ( ) 0=f c , then c is called a zero of f, or a

root of f. If c is a zero of f, then

• c is an x-intercept of the graph of f. • ( )−x c is a factor of f.

So if we have a polynomial in factored form, we know all of its x-intercepts.

• every factor gives us an x-intercept. • every x-intercept gives us a factor.

Example 1: Find the zeros of the polynomial: 3 2( ) 5 6= − +P x x x x Example 2: Consider the function 4 3 2( ) 3 ( 3) (5 2)(2 1) (4 )= − − − − −f x x x x x x .

Zeros (x-intercepts): To get the degree, add the multiplicities of all the factors: The leading term is:

M 1310 4.1 Polynomial Functions 5 Behavior at Intercepts: Near an x-intercept, c, the shape of the function is determined by the factor (x-c) and the power to which it is raised. Let’s look again at the graph on the first page

3 2( ) ( 2) ( 1)= − +f x x x x

Notice the shape of the graph as it crosses the x-axis at each intercept. Definition: The multiplicity of a factor in a polynomial function is the power to which it is raised in the fully factored form of the polynomial. The multiplicity of the factor (x-c) with no higher power is 1. Compare the multiplicities of the factors in the above polynomial with the shape at each corresponding intercept. The graph of ( ) ( )= − kP x x c looks like the function ( ) = kf x x near the x-intercept c. Since we know the shape of a power function, we have the following rules:

• Even multiplicity: touches x-axis, but doesn’t cross (looks like a parabola there).

• Odd multiplicity of 1: crosses the x-axis (looks like a line there).

• Odd multiplicity 3≥ : crosses the x-axis and looks like a cubic there.

M 1310 4.1 Polynomial Functions 6 Steps to graphing other polynomials: 1. Find the y-intercept by finding P(0).

2. Factor and find x-intercepts. 3. Mark x-intercepts on x-axis.

4. For each x-intercept, determine the behavior.

• Even multiplicity: touches x-axis, but doesn’t cross (looks like a parabola there).

• Odd multiplicity of 1: crosses the x-axis (looks like a line there).

• Odd multiplicity 3≥ : crosses the x-axis and looks like a cubic there.

Note: It helps to make a table as shown in the examples below. 5. Determine the leading term.

• Degree: is it odd or even?

• Sign: is the coefficient positive or negative? 6. Determine the end behavior. What does it “look like”?

Odd Degree Odd Degree Even Degree Even Degree Sign (+) Sign (-) Sign (+) Sign (-) 7. Draw the graph, being careful to make a nice smooth curve with no sharp corners. Note: without calculus or plotting lots of points, we don’t have enough information to know how high or how low the turning points are.

M 1310 4.1 Polynomial Functions 7 Example 3: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if

possible and have correct end behavior.

( ) ( )3 24( ) 2 1= − +P x x x x

Example 4: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if

possible and have correct end behavior.

( ) ( )23( ) 2 3= + −P x x x x

M 1310 4.1 Polynomial Functions 8 Example 5: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if

possible and have correct end behavior.

( )2( ) 2 1 ( 3)= − + −P x x x

Example 6: Find the zeros then graph the polynomial. Be sure to label the x intercepts, y intercept if

possible and have correct end behavior. 3 2( ) 3 4 12= + − −P x x x x

M 1310 4.1 Polynomial Functions 9 Example 7: Given the graph of a polynomial determine what the equation of that polynomial.

Example 8: Given the graph of a polynomial determine what the equation of that polynomial.