Name:
Teacher:
Period:
Due Date: 10/28 (2.1A), 10/29 (2.1B), 10/30 (2.1C)
WHAT IS A POLYNOMIAL?
A polynomial can have:
 Constants
 Variables
 Whole Number Exponents
A polynomial can’t have:
 Negative exponents
 Division by a variable
 Fractional exponents
PRE CALCULUS HW 2.1 ABC
TURNING POINTS OF A POLYNOMIAL FUNCTION
A polynomial function f of degree n≥1 has at most n-1
turning point.
A parabola has a degree of 2 and has at most 1 turning
point.
Examples of polynomials are 5, 3x, 4x2+5x-3
LEADING CO-EFFICIENT
The leading co-efficient is the co-efficient of the variable
with the greatest exponent.
The leading co-efficient of 4x2+5x-3 is 4.
The function below has a degree of 3 and has at most 2
turning points.
DEGREE OF A POLYNOMIAL FUNCTION
The degree of a polynomial function is equal to the
greatest exponent.
The degree of f(x)=4x2+5x-3 is 2.
The parity of the degree is whether the degree is even or
odd.
END BEHAVIOR OF POLYNOMIAL FUNCTIONS
For a polynomial function with a leading coefficient k, the
chart below describes the end behavior (as x approaches
either infinity) of functions with even or odd exponents.
DISTINCT REAL ZEROES OF A POLYNOMIAL FUNCTION
A polynomial function f of degree n≥1 has at most n
distinct real zeroes. No picture. Distinct means different
and you should know what zeroes are.
Y-INTERCEPT OF A POLYNOMIAL FUNCTION
Every polynomial function has 1 and only 1 y-intercept.
MULTIPLICITY
If there is a zero of a polynomial function that has an
exponent like (x-c)m, then m is the multiplicity of the zero
c.
If m is even, the polynomial function is tangent to the x-axis
at point c.
If m is odd, the polynomial function crosses the x-axis at
point c.
Name:
Teacher:
Period:
Due Date: 10/28 (2.1A), 10/29 (2.1B), 10/30 (2.1C)
COMPLETE HW IN YOUR NOTEBOOK (WITH PLENTY OF
ROOM TO EDIT)
HW 2.1A
Complete the following problems for Wednesday, Oct 28th
1. Describe the end behavior of
𝑓(𝑥) = −2𝑥 3 + 5𝑥 2 − 7𝑥 + 2 using limits and
using the leading co-efficient and parity of degree.
2. State the number of distinct real zeroes and
turning points of 𝑔(𝑥) = −6𝑥 7 + 25.
3. Determine the real zeroes of 𝑓(𝑥) = 𝑥 4 − 3𝑥 2 − 4
4. State the multiplicity of each zero of
𝑓(𝑥) = 6𝑥(𝑥 − 5)(𝑥 + 2)3 (𝑥 − 1)4
5. Sketch a function that is tangent to the x-axis at
x = 5.
6. Sketch a function that crosses the x-axis at x = 5.
7. Describe everything you know that is different
about two polynomial functions if one is described
in #4 and the other is described in #5.
8. Based on the facts concerning what a polynomial
can have and can’t have, what can’t possibly
happen in a polynomial function? How would you
describe a polynomial function?
1.
2.
3.
4.
5.
6.
HW2.1B
Determine the real zeroes of
𝑓(𝑥) = 𝑥 4 − 9𝑥 2 + 18
Determine the real zeroes of
𝑔(𝑥) = 𝑥 5 − 6𝑥 3 − 16𝑥
Describe the end behavior of 𝑔(𝑥) = 4𝑥 6 + 2𝑥 + 7
using limits and using the leading co-efficient and
parity of degree.
Determine the real zeroes and the multiplicity of
each real zero of 𝑓(𝑥) = 3𝑥 5 − 18𝑥 4 + 27𝑥 3
Sketch 𝑔(𝑥) = 5(𝑥 − 1)(𝑥 + 2)3 (𝑥 + 6)2
Sketch 𝑓(𝑥) = 3𝑥 3 − 3𝑥 2 − 36𝑥
PRE CALCULUS HW 2.1 ABC
HW2.1C
For these functions:
1
1. 𝑓(𝑥) = 8𝑥 4 − 𝑥 2 + 𝑥
2. 𝑔(𝑥) = −5𝑥 7 + 6𝑥 4 + 8
3. ℎ(𝑥) = 3𝑥 6 + 4𝑥 3 − 3𝑥 − 8
4. 𝑚(𝑥) = 5𝑥 2 + 8𝑥 5 − 𝑥 −3
1
5. 𝑛(𝑥) = 5𝑥 4 + 𝑥 5 − 9
3
Answer the following questions (if a function is not a
polynomial, you only answer question a):
a. Is this a polynomial? Explain why or why
not?
b. What is the degree of the polynomial?
c. What is the leading coefficient of the
polynomial?
d. What are the possible number of distinct
real zeroes?
e. Determine the end behavior using the
leading co-efficient and parity of degree.
6. Determine the zeroes, state the multiplicity of
each zero, and describe the behavior of the
function at each zero.
a. 𝑓(𝑥) = −(𝑥 + 2)2 (𝑥 − 4)2
b. 𝑓(𝑥) = −2𝑥 3 − 4𝑥 2 + 6𝑥
c. 𝑓(𝑥) = 𝑥 6 − 6𝑥 3 − 16