Download MATH 245CHAPTER 3 WORKSHOP

MATH 245
CHAPTER 3
WORKSHOP
1. Sketch the graph f  x    x  4  3
2
2. Graph the quadratic function. Give (a) vertex (b)axis (c) domain (d) range (e) determine the
interval of the domain for which the function is increasing and decreasing
a)
f x   x 2  10 x  21
b)
f x   2 x 2  12 x  16
3. Use synthetic division to perform the division
x 4  4x3  2x 2  9x  4
x4
4. Use the remainder theorem and synthetic division to find f(k)
f x   2 x 5  10 x 3  19 x 2  50; k  3
5. Use Synthetic division to decide whether the given number k is a zero of the given polynomial
function. If it is not, give the value of f(k)
a)
f x   2 x 3  9 x 2  16 x  12; k  1
b)
f x   x 2  3x  5; k  1  2i
6. Use the factor theorem and synthetic division to decide whether the second polynomial is a
factor of the first 2 x 4  5 x 3  8 x 2  3x  13; x  1
7. Factor f(x) into linear factors given that k is a zero of f(x).
f x   6 x 3  13x 2  14 x  3
3
2
8. For the polynomial function f x   x  x  4 x  6; 3
3
2
9. For the polynomial function f x   x  6 x  x  30 (a) list all possible rational zeros (b)find
all rational zeros and (c) factor f(x)
10. Find a polynomial function f(x) of degree 3 with real coefficients that satisfies the given
conditions
a) Zeros of -3, 1 and 4; f(2)=30
b) Zeros of 0 and zero of 1 having multiplicity 2; f(2)=10
11. Find a polynomial function f(x) of least degree having only real coefficients with zeros as given
2 and 3 + i
12. Use Descartes’ rule of signs to determine the different possibilities for the number of positive,
3
2
negative and nonreal complex zeros for the function f x   4 x  x  2 x  7
13. Sketch the graph of the polynomial function f  x    x  1  1 . Determine the intervals for
3
the domain for which the function is increasing or decreasing.
14. Graph the polynomial function. Factor first if the expression is not in factored form.
a)
f x   x 2 x  5x  3x  1
b)
f x   x 3  5 x 2  x  5
15. Use the intermediate value theorem for polynomials to show the polynomial function has a real
zero between the numbers given
f x   2 x 4  4 x 2  4 x  8; 1 and 2
16. Show the real zeros of each polynomial function satisfy the given conditions
a)
f x   2 x 5  x 4  2 x 3  2 x 2  4 x  4; no real zero greater than 1
b)
f x   x 5  2 x 3  2 x 2  5x  5; no real zero less than -1
17. Explain how the graph of the function can be obtained from the graph of y 
1
. Then graph f
x
and give the (a) domain and (b) range. Determine the intervals of the domain for which the
function is (c) increasing or (d) decreasing
f x  
1
x2
18. Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each
rational function
a)
b)
c)
6
x9
4  3x
f x  
2x  1
f x  
x2  4
f x  
x 1
19. Sketch the graph of the rational function
a)
b)
x3
x4
2x  1
f x   2
x  6x  8
f x  
c)
f x  
x  12
x  2x  3
d)
f x  
2x 2  3
x4
20. A toy rocket is launched straight up from the top of a building 50 ft tall at an initial velocity of
200 ft per sec.
a) Give the function that describes the height of the rocket in terms of time t.
b) Determine the time at which the rocket reaches its maximum height and the maximum
height in feet.
c) For what time interval will the rocket be more than 300 ft above ground level?
d) After how many seconds will it hit the ground?