Aim How do we write a polynomial function
- Slides: 6
Aim: How do we write a polynomial function with given roots? Do Now: Given the graph of two functions, write the equations in factored form for both
Although both functions have the same zeros, the function must be different according to the graphs We can only write the equations based on the graph not the original function How do we write the functions? Fundamental Theorem of Algebra: Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. Every polynomial �� (�� ) of degree �� , �� > 0, can be written as a product of a constant �� , �� ≠ 0, and �� linear factors: �� (�� ) = �� (�� − �� ) 1)(�� 2)⋯(�� �� A polynomial equation of degree �� has exactly �� complex roots, namely �� 1, �� 2, �� 3⋯�� ��
The graph below is of a third-degree polynomial function f. a. State the zeros of f. b. Write a formula for f in factored form using c for the constant factor. c. Use the fact that f(-4) = -54 to find the constant factor c. d. Verify your equation by using the fact that f(1)= 11.
Consider the graph of a degree 5 polynomial shown below, with x-intercepts, −�� , and �� a) Write a formula for a possible polynomial function that the graph represents using �� as the constant factor. b) Suppose the y-intercept is −��. Find the value of �� so that the graph of P has y-intercept −��.
For each of the following, write a polynomial function with least degree whose roots are given.
Modeling with Polynomial Functions For a fundraiser, members of the math club decide to make and sell “Pythagoras may have been Fermat’s first problem but not his last!” t-shirts. They are trying to decide how many t-shirts to make and sell at a fixed price. They surveyed the level of interest of students around school and made a scatterplot of the number of t-shirts sold (x) versus profit shown below. a) Identify the y-intercept. Interpret its meaning within the context of this problem. b) If we model this data with a function, what point on the graph of that function represents the number of t-shirts they need to sell in order to break even? Why? c) How many t-shirts should they sell in order to maximize the profit? d) What is the maximum profit?
