In our study of polynomial functions with rational coefficients, we learned that irrational and nonreal solutions must occur in conjugate pairs. In addition, the factor theorem tells us that if x = c is a zero, then x – c is a factor.Use this information to complete Exercises 1 and 2.
1. A function f(x) with rational coefficients has the 2. A function g(x) with rational coefficients has the
zeroesx = ,x = 4 and x = –5. (a) Write the zeroes x = 2, x = –1 and x = 2i. (a) Write the
function in completely factored form if it must function in completely factored form if it must
be of minimum degree. Then (b) writethe be of minimum degree. Then (b) write the
polynomial in standard form. Assume a = 1. function in standard form. Assume a = 1.
Use synthetic division and the quotient polynomial, along with the factor theorem, to complete Exercises 3 and 4.
3. Use synthetic division to show that x = 2 is 4. Use synthetic division to show that x = –4 is
a zero of f(x) = x3 – 3×2 – 10x + 24, then use a zero of g(x) = x3 – 13x + 12, then use
theresult to write f in completely factored the result to write g in completely factored
form and name the other zeroes. form and name the other zeroes.
In our study of polynomial functions with rational coefficients, we learned that a polynomial of degree n> 1 will have exactly n zeroes, which may be a combination of real, nonreal, and/or repeated zeroes. Further, we know that the real zeroes appear graphically as x-intercepts. Use this information to complete Exercises 5 and 6.
5. The graph of a function f is given. Use the 6. The graph of a function g is given. Use the
function and its graph to state the number of function and its graph to state the number of
real zeroes and the number of nonreal zeroes real zeroes and the number of nonreal zeroes
for this function. for this function.
f(x) = –0.25×3+ 3x + 5 g(x) = 2×5 – 7×3 + 4x – 2
a. real zeroes:00_00 a. real zeroes:00_00
b. nonreal zeroes:00_00 b. nonreal zeroes:00_00
The graph shown represents the amount of water in a reservoir that supplies water to a large metropolitan area. Here, W(t) represents the water level over a six-month period in millions of gallons above or below normal, for time t in months. Use this information to complete Exercise 7.
7. For the graph of W(t) shown:
a. What is the minimum possible degree of the polynomial
that could model this graph?
b. How many months was the water level below normal
during this six-month period?
c. Approximately how many gallons above or below normal were in the reservoir in month 5?
d. Use the zeroes of the function to construct a polynomial model in factored form.Be sure to adjust
the leading coefficient using the point (3, 12) as needed.
While not the primary focus of this course, the factor and remainder theorems can also be applied using nonreal zeroes. Use this information along with synthetic division to complete Exercises 8 and 9.
8. Use synthetic division to show that x = 2i 9. Use synthetic division to show that x = –3i
isa zero of f(x) = x3+3×2+4x + 12. isa zero of g(x) = x3 – 5x² + 9x – 45.