1. Chapter 2 – Polynomial and Rational Functions 2.2 – Polynomial Functions of Higher Degree

2. Properties of Polynomial Functions Graphs of polynomial functions are smooth and continuous The domain is all real numbers End behavior: 1.The leading coefficient (a) will tell you if the graph finishes high or low 2.The degree of the function will tell you if the graph starts where it finishes  Even-degree functions finish where they start  Odd-degree functions finish opposite of where they start a > 0 a < 0 Odd degree Even degree

3. Without a calculator, decide which function represents the graph below: 1. 2. 3. 4.

4. Without a calculator, describe the right- and left-hand behavior of the following function: 1. Starts low, finishes high 2. Starts low, finishes low 3. Starts high, finishes low 4. Starts high, finishes high

5. Zeros of a polynomial For a polynomial f of degree n… ◦…f has at most n real zeros. ◦…f has at most (n-1) extrema (relative minima or maxima) If f is a polynomial and a is a real number, the following statements are equivalent: 1.x = a is a zero of the function f. 2.x = a is a solution of f(x) = 0. 3.(x – a) is a factor of f(x). 4.(a, 0) is an x-intercept of the graph of f.

6. Zeros of a polynomial Ex: Find all real zeros of f(x) = 6x 4 – 33x 3 – 18x 2. ◦Factor! Don’t forget to factor out any common factors immediately! ◦3x 2 (2x 2 -11x – 6) ◦3x 2 (2x 2 – 12x + x – 6) ◦3x 2 (2x + 1)(x – 6) ◦So the zeros are at x = -½, 0, and 6. ◦Graph to confirm! Use the calc/zero functions! Note: In the last example, x = 0 is a repeated zero with a multiplicity of 2 since 3x 2 = 0. ◦Even multiplicity – the graph bounces off that x-intercept ◦Odd multiplicity – the graph goes through that x-intercept

7. Zeros of a polynomial Ex: Find a polynomial of degree 3 that has zeros at x = -1/3, 0, and 2. ◦Write it as factors, then multiply it out! ◦Try to avoid fractions if you can ◦f(x) = (3x + 1)(x – 0)(x – 2) ◦f(x) = x(3x + 1)(x – 2) ◦f(x) = x(3x 2 + x – 6x – 2) ◦f(x) = x(3x 2 – 5x – 2) ◦f(x) = 3x 3 – 5x 2 – 2x If you need to find a polynomial of degree 4 that has those same 3 zeros, just multiply f by (3x + 1), x, or (x – 2)! ◦Ex: f(x) = 3x 4 – 5x 3 – 2x 2

8. Zeros of a polynomial Ex: Find a polynomial of degree 3 that has zeros at ◦Write it as factors, then multiply it out! ◦f(x) = x(x – ( ))(x – ( )) ◦f(x) = x(x – 2 – )(x – 2 + ) ◦f(x) = x((x – 2) 2 – 2 ) ◦f(x) = x(x 2 – 4x + 4 – 11) ◦f(x) = x(x 2 – 4x – 7) ◦f(x) = x 3 – 4x 2 – 7x

9. Find a polynomial of degree 4 with the following zeros: x = 2, 0, -2, 3 (Don’t use a calculator!) 1. 2. 3. 4.

10. Graphing Polynomials Ex: Graph f(x) = x 3 + x 2 – 2x by hand. 1.Check end behavior  It rises to the right and falls to the left! 2.Find zeros Factor if possible! f(x) = x(x 2 + x – 2) f(x) = x(x – 1)(x + 2) x–ints = 0, 1, -2 3.Plot a few other points 4.Graph it! x½1.5 f(x)2-.6252.625