Polynomial Functions A polynomial of degree n is a function of the form P(x) = anxn + an-1xn-1 + ... + a1x + a0 Where an 0. The numbers a0, a1, a2, . . . , an are called the coefficients of the polynomial. The a0is the constant coefficientorconstant term. The number an, the coefficient of the highest power, is the leading coefficient, and the term anxn is the leading term.
Graphs of Polynomial Functions and Nonpolynomial Functions
Graphs of Polynomials Graphs smooth curve Degree greater than 2 ex. f(x) = x3 These graphs will not have the following: Break or hole Corner or cusp Graphs are lines Degree 0 or 1 ex. f(x) = 3 or f(x) = x – 5 Graphs are parabolas Degree 2 ex. f(x) = x2 + 4x + 8
Even- and Odd-Degree Functions
The Leading-Term Test
Finding Zeros of a Polynomial Zero- another way of saying solution Zeros of Polynomials Solutions Place where graph crosses the x-axis (x-intercepts) Zeros of the function Place where f(x) = 0
Using the Graphing Calculator to Determine Zeros Graph the following polynomial function and determine the zeros. Before graphing, determine the end behavior and the number of relative maxima/minima. In factored form: P(x) = (x + 2)(x – 1)(x – 3)²
MultiplicityIf (x-c)k, k 1, is a factor of a polynomial function P(x) and: K is even The graph is tangent to the x-axis at (c, 0) K is odd The graph crosses the x-axis at (c, 0)
Multiplicity y = (x + 2)²(x − 1)³ Answer. −2 is a root of multiplicity 2, and 1 is a root of multiplicity 3. These are the 5 roots: −2, −2, 1, 1, 1.
Multiplicity y = x³(x + 2)4(x − 3)5 Answer. 0 is a root of multiplicity 3, -2 is a root of multiplicity 4, and 3 is a root of multiplicity 5.
To Graph a Polynomial Use the leading term to determine the end behavior. Find all its real zeros (x-intercepts). Set y = 0. Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph. Find the y-intercept. Set x = 0. Use any additional information (i.e. turning points or multiplicity) to graph the function.