## Math 116: Study Guide Chapter 4

1. Sketch the graph of the given functions. Pay special attention to the x-intercepts and the left and right hand behavior. Four parts. (1 point for having the proper behavior at each x-intercept, 1 point for having the proper behavior on the right hand side). (24 points total).
2. Write a polynomial function in factored form which exhibits the same behavior as the graph shown. (1 point each for the correct right hand behavior and the behavior at each x-intercept). (4 points total).
3. Find a polynomial function with integer coefficients which has the given roots. Remember that complex roots and square roots come in pairs. You don't have to multiply it out completely, but make sure all of the coefficients are integers (you will have to multiply out any complex roots or square roots to make the coefficients integers). Three parts. (9 points total). Look at problems 4.5.27 - 4.5.36.
4. Use synthetic division to evaluate a polynomial at a specific point. (4 points). Look at problems 4.3.45 - 4.3.50.
5. Given a polynomial function, list all possible combinations of positive, negative, and complex roots, using Descartes' rule of signs. List all possible rational zeros. (8 points). Look at problems 4.4.1 - 4.4.16.
6. You are given a value and a possible zero. State either the upper bound theorem or the lower bound theorem (see page 286). Which one you state is dependent upon the value given. Remember that upper bound is appropriate if the value is positive and lower bound is appropriate if the value is negative. Then, determine if the value given is actually the bound you said was appropriate. (6 points)
7. Consider a polynomial function. Given one of the roots, find the remaining roots. Then completely factor the polynomial using linear and irreducible quadratic factors. Look at problems 4.5.37 - 4.5.50. (6 points).
8. State the Fundamental Theorem of Algebra. (3 points)
9. Consider the general form a polynomial in one variable of degree n (see page 247). Six parts (12 points). You need to fill in the blank with an expression involving portions of the function. For example, the y-intercept is f(0) which is a0.
1. Determine which part of that polynomial determines the right hand behavior (see page 260),
2. left hand behavior (see page 260),
3. possible rational zeros (see page 283),
4. y-intercept,
5. the maximum number of positive and negative real roots (see page 281),
6. the maximum number of turns in the graph (see page 261).
10. Determine whether the graph of a function will cross the x-axis or touch the x-axis (see page 262). This has something to do with the multiplicity of the factor. (4 points)

#### Notes:

• None of the problems are directly from the text.
• The in-class portion of this test is 80 points.
• The take-home portion of this test is 20 points.