Practice Problems for Module 3 Unit #1

Decide is the given function is a polynomial function. In case it is, write it
in standard form and state its degree. If it is not, explain why.
(Refers to Example A)
1.
f ( x)  3x  2 x 2  6 x 1  4 Answer: not a polynomial since the third
term contains x to a negative power.
f ( x)  23  x  x 3  2 x 2
2.
Answer: this is a polynomial, all powers
of x are positive integers, there is also a constant term. The coefficient
2 is a real coefficient. Its degree is 3 and its standard form:
f ( x)   x3  2 x 2  x  23
3.
Answer: Not a polynomial, since the third
f ( x)  5 x 4  2 x 2  3 x 2  5
term contains an x to a fractional power.
1
4.
5.
x 2  3x  8
Answer: Not a polynomial since there is an x
x
appearing in the denominator (equivalent to negative exponent).
f ( x) 
f ( x)  2 x  3 x 2  7 x  1 Answer: Not a polynomial since the first term
contains
6.
7.
1
2
x which corresponds to x (fractional exponent).
2 x2
x
 5 x3  8 
Answer: This is a polynomial. The only
3
5
fractions that appear involve just numbers (not variables) which can be
regarded as rational coefficients. Its degree is 3 and its standard form:
2x2 x
3
f ( x)  5 x 
 8
3
5
f ( x)  
f ( x)  5 x3 
2
  x4  4x
Answer: This is a polynomial. The

coefficients contain expressions with the number  , but they are real
numbers. Its degree is 4 and its standard form:
f ( x)   x 4  5 x 3  4 x 
2

8.
f ( x)  2 x 4  3x  3x 2  2 Answer: Not a polynomial. The second
term contains x to the power  , which is an irrational number (not a
positive integer).
9.
f ( x)  5 Answer: This is a polynomial, which consists of a
numerical term. Its degree is zero, and its standard form: f ( x)  5
1
Answer: Not a polynomial since the last
x
term contains an x in the denominator (equivalent to a negative
exponent).
10. f ( x)  2 x 4  x 3  2 x  5 
11. f ( x)  2  5 x Answer: This is a polynomial consisting of a linear
term and a constant term. Its degree is 1 and its standard form:
f ( x)  5 x  2
12. f ( x)  2
Answer: This is a polynomial of degree 0, just a numerical
term with real coefficient. Its standard form: f ( x)  2

A list of polynomials id given below. Write the expressions of their
“leading terms”. Note that the polynomials are not in standard form.
f ( x)  9 x 2  4 x  5x3  2 x7  3
16.
Answer: the leading term is: 2x 7
Answer: the leading term is: 3x 4
f ( x)   x 2  5  3 x 4  5 x 3  x
1
3
3
f ( x)   x 2  6 x  x 6  8 x 5  1 Answer: the leading term is:  x 6
2
4
4
5
2
5
3
f ( x)  10  3x  x  7 x  x Answer: the leading term is:  x
17.
f ( x)  4 x  20  3x 4  3x 2  15 x 3
13.
14.
15.

18.
19.
Answer: the leading term is:
3x 4
What is the “leading coefficient” of each of the following polynomials?
f ( x)  4 x 2  2 x  8x3  3x7  7
Answer: the leading coefficient is: 3
Answer: the leading coefficient is: -1
21.
f ( x)  2 x  1  x  6 x  4 x
3
1
1
f ( x)   x 2  2 x  x 6  5 x 5  4 Answer: the leading coefficient is: 
8
4
4
6
2
5
3
f ( x)  x  5  2 x  6 x  10 x  4 x Answer: the leading coefficient is: 1
22.
f ( x)  x  3  5 x 4  5 x 2  8 x 3
23.
f ( x) 
20.
2
4
3
2 x  7  4 x 4  3x 2  8 x3
2
Answer: the leading coefficient is:
Answer: the leading coefficient is: 2
5

A polynomial function f ( x) is given below.
1
f ( x)  3 x 2  x  6 x 3  x 5  4
3
Identify all of the following:
(Refers to Example B)
24.
25.
26.
degree of the polynomial
leading term
leading coefficient
27.
linear term
28.
29.
30.
constant term
quadratic term
cubic term

Answer: 5
Answer:  x 5
Answer: -1
1
Answer: x
3
Answer: 4
Answer: 3x 2
Answer: 6x 3
A polynomial function f ( x) is given below.
1 x4
f ( x)   5 x 3    4 x 2
2 3
Identify all of the following:
(Refers to Example B)
31.
degree of the polynomial
32.
leading term
33.
leading coefficient
34.
linear term
35.
constant term
36.
37.
quadratic term
cubic term

Answer: 4
1
Answer: x 4
3
1
Answer:
3
Answer: 0x
1
Answer: 
2
Answer: 4x 2
Answer:  5x3
A polynomial function f ( x) is given below.
f ( x) 
6 x  5 x 4  x 6  4 x5  4 x 2
3
Identify all of the following:
(Refers to Example B)
38.
degree of the polynomial
39.
leading term
40.
leading coefficient
41.
42.
linear term
constant term
43.
quadratic term
44.
cubic term
45.
Answer: 6
1
Answer:  x 6
3
1
Answer: 
3
Answer: 2x
Answer: 0
4
Answer:  x 2
3
Answer: 0x3
The graph of a power function is given below. Decide if it
corresponds to an even integer power or to an odd integer power.
Justify your answer.
Answer: An odd positive integer power function, since it looks like a “snake”,
increasing with x and going through the point (0,0).
46.
The graph of a power function is given below. Decide if it
corresponds to an even integer power or to an odd integer power.
Justify your answer.
Answer: An even positive integer power function, since it looks like a “cup”, with
branches upward, and “touching” the x axis at x=0.
47.
The graph of a power function is given below. Decide if it
corresponds to an even integer power or to an odd integer power.
Justify your answer.
Answer: An odd positive integer (one), the graph is that of a line (linear function)
going through (0,0). The slope is negative indicating that the coefficient in the
linear power function is a negative number.
48.
The graph of a power function is given below. Decide if it
corresponds to an even integer power or to an odd integer power.
Justify your answer.
Answer: An even positive integer power, since the graph looks like a cup which
“touches” the x-axis at x=0. The branches pointing downward indicate that the
coefficient in the power function is a negative number.
49.
The graph of a power function is given below. Decide if it
corresponds to an even integer power or to an odd integer power.
Justify your answer.
Answer: An odd positive integer power. The shape of the graph is that of a
“snake” passing through (0,0). The graph is decreasing, which indicates that the
coefficient in the power function is negative.
50.
The graph of a function is given below. Decide if it could be the
graph of a power function. Justify your answer.
Answer: Yes, it could be the graph of an even positive integer power function.
The graph has the shape of a cup, “touching” the x-axis at x=0. The branches
pointing down indicate a negative coefficient in the power function.
13.- The graph of a function is given below. Decide if it could be the graph of a
power function. Justify your answer.
Answer: No, this is not the graph of a power function. The graph looks like a
“snake”, but it is not going through the origin of coordinates (0,0), so it is not a
power function.
51.
The graph of a function is given below. Decide if it could be the
graph of a power function. Justify your answer.
Answer: No, this is not the graph of a power function. The graph shows a “cup”
shape, but it is not “touching” the x-axis at x=0, so it is not a power function.
52.
The graph of a function is given below. Decide if it could be the
graph of a power function. Justify your answer.
Answer: No, this is not the graph of a power function. This is the graph of a
horizontal line, which means that there is no power of x in the functional
expression (the slope is zero), so it is not a power function.