Algebra II Unit 4 Homework
Homework 4.1
1. How can you identify the degree of any given polynomial?
2. Give an example of each of the following polynomials.
a. Quartic binomial
b. Quadratic monomial
c. Cubic polynomial with four terms
3. What is the standard form of a polynomial expression? What do all of the different a’s represent?
4. Let f(x) = 8x3 – 2x2 + 4x + 5. Find each of the following
a. f(3)
c. f(3) + f(-1)
b. f(-1)
d. f(f(0))
5. Give an example of a function that is not a polynomial.
Homework 4.2
1. When setting up a synthetic substitution problem, how many lines should you draw? For example,
how many lines should you draw for this polynomial: f(x) = 6x7 – 8x2 + 3x – 2.
2. What is the theorem stated in this section of the notes?
3. According to the theorem above, if f(9) = 0 for some given polynomial, what does that tell you
about (x – 9) in relationship to that polynomial? On the other hand, if f(8) = 7 (or any number
other than 0), what does that tell you about (x – 8) in relationship to that polynomial?
4. State two connections between synthetic substitution and long division of polynomials (Note: one
connection should have to do with the remainder of the long division problem and the other
should have to do with the quotient).
5. Let f(x) = x4 – 8x3 – 11x2 – 12x + 270. Use synthetic substitution to find each of the following.
a. f(3)
c. f(3) + f(4)
b. f(4)
d. f(f(3))
Homework 4.3
1. List the first 10 perfect squares, then list the first 10 perfect cubes.
2. Give these three formulas:
a. Difference of Squares
b. Difference of Cubes
c. Sum of Cubes
3. Give two examples of difference/sum of cubes problems. Make one an expression and make the
other an equation.
4. List the steps for factoring a sum/difference of cubes binomial.
5. State whether each of the following can be a sum/difference of cubes problem or not (in your
journal, write the problem, then write “yes” or “no”):
a. 2x3 + 54
d. x6 – 125
b. x2 – 64
e. x3 – 3x2
3
c. 5x – 27
f. 4x3 – 16x2 – 8
Homework 4.4
1. A student attempted to solve the following polynomial equation like this:
2x4 – 18x2 = 0
2x2(x2 – 9) = 0
2x2(x – 3)(x + 3) = 0
Zero Product Property: (x – 3) = 0 and (x + 3) = 0
x = 3 and x = -3 are the only solutions.
2.
3.
4.
5.
The student above missed something. Find their error and fix it.
Use information from section 1.1 and 3.4 to answer this question. What is a rational root?
Use the rational root theorem to list all of the possible rational roots of the following functions:
a. f(x) = x4 + 2x2 – 24
b. f(x) = 2x3 + 9x2 – 53x – 60
A student is trying to find all of the rational roots of this function: f(x) = 3x4 + 2x3 – x + 15
She says the possible roots are: 1, 3, 15, 1/3, 5/3. What did she leave out?
Let f(x) = x6 – 2x4 – 11x2 + 12. Is x = 1 a root of the function? Explain why or why not.
Homework 4.5
Explain when you would use each of the five factoring methods given below. Then give an example.
1.
2.
3.
4.
5.
Difference of Squares
U-Substitution
Sum/Difference of Cubes
Factor by Grouping
Synthetic Substitution
Homework 4.6
1. What is the name of a method that can be used to multiply polynomials?
2. What size box should you make in order to multiply the polynomials (x3 – 4x2 + 5x – 1)(3x – 2)?
3. Explain in your own words, what are the roots of a polynomial? Then explain the relationship
between the roots of a polynomial and the factors of a polynomial.
4. A certain polynomial has these roots: x = 3, x = ½, x = -½, and x = -5. What is the factored form of
that polynomial?
5. A student factored the polynomial f(x) = 3x3 – 4x2 – 61x + 20 and got (3x – 1)(x + 4)(x – 5). Check
their answer.
Homework 4.7
1. Describe the end behavior of the graph of the polynomial: y = 4x5 – 8x2 + 3x – 6.
2. If you are given a polynomial function, how would you find the y-intercept of its graph?
3. “Zeroes” is another name for the “roots” of a function. When graphing a polynomial, what is the
difference between the behavior of the graph at a zero with an even multiplicity and an odd
multiplicity?
4. Describe the process you would have to undergo to find the zeroes of this polynomial: y = x4 – 5x2
+ 4.
5. What is the number of turning points for the graph of the polynomial y = -3x4 – 6x + 2?