Polynomial Functions and their Graphs Section 2.3: Polynomial Functions and their Graphs What you’ll learn Identify polynomial functions. Recognize characteristics of graphs of polynomial functions. Determine end behavior. Use factoring to find zeros of polynomial functions. Identify zeros and their multiplicities. Use the intermediate value theorem. Understand the relationship between degree and turning points. 8. Graph polynomials functions.

Examples of polynomials functions: Degree 5 Degree 6 Not Polynomial Functions. The exponent in the variable is not an integer The exponent on the variable is not a nonnegative integer

Smooth, Continuous Graphs End Behavior of Polynomial Functions The degree of a polynomial function affects the shape of its graph and determines the maximum numbers of turning points. TURNING POINTS are the points where the graphs changes directions. It also affects the end behavior. END BEHAVIOR is the directions of the graph to the far left and the far right. End behavior of a Polynomial Function of Degree n with Leading Term n even n odd a positive up and up down and up a negative down and down up and down

Odd-degree polynomial functions have graphs with opposite behavior at each end. Even-degree polynomial functions have graphs with the same behavior at each end. Answer Leading coefficient is 1 (+) and the degree is 3,so is odd Answer 3+2+1=6 even Leading coefficient is -4 and the degree is 6,so is even

Zeros of Polynomial Functions Answer Leading coefficient 1 (+) Degree 3: odd Try the G.C.

Answer Make the equation =0

Answer Because the multiplicity of -1 is odd, the graph crosses the x-axis at this zero. Because the multiplicity of 3/2 is even, the graph touches the x-axis and turns at this zero.

The Intermedia Value Theorem Use G.C. use ZERO feature. Answer Since f(2)=-1 and f(16), the sign change shows that the polynomial function has a zero between 2 and 3. Take a note: If a f is a polynomial function of degree n, then the graph of f has at most n-1 turning points.

Steps to complete a polynomial graph.

Answer The leading coefficient,1, is positive, the degree of the polynomial function,4 is even

The partial graph or information we have suggests y-axis symmetry. Let’s verify this by finding f(-x). the graph is symmetric with respect to the y-axis. Because n=4 and 4-1=3 The graphs has three turning points, which is the maximum number for a fourth degree polynomial function. Verify that 1 is a relative maximum and(0,1) is turning point.

Let’s remember the procedure for dividing one polynomial by another.

Answer _ _ _ _

Answer Take a note: If a power of x is missing in either a dividend or a divisor, add that power of x with a coefficient of 0 and then divide. In this way, like terms will be aligned as you carry out the long division. + + + Remainder

Dividing Polynomials Using Synthetic Division.

Answer The divisor must be in the form . Thus, we write as . This means that . Writing a 0 coefficient for the missing Term the dividend, we can express the division as follow

The Remainder Theorem. Answer 2 -4 2 1 -2 1 5 By the Remainder Theorem, if f(x) is divide by x-2, then the remainder is f(2). We’ll use synthetic division to divide. The remainder,5, is the value of f(2)=5. We can verify that this is correct by evaluating f(2) 2 -4 2 1 -2 1 5

The Factor Theorem Answer 3 2 -3 -11 6 6 9 -6 2 3 -2 3 2 -3 -11 6 6 9 -6 2 3 -2 The remainder,0, verifies that x-3 is a factor of

Using G.C.