Algebra 2 Advanced – Polynomial Functions Assessment
Name: ___________________
Part 1 – Fact recollection
1) Given 𝑃(𝑥) = −2𝑥 6 − 5𝑥 5 + 7𝑥 − 3
(a) State the leading term
(b) State the leading coefficient
(c) Sate the degree of the polynomial
(d) What is the end behavior of 𝑃(𝑥)?
2) Given 𝑃(𝑥) = 3𝑥 5 (2𝑥 − 1)2 (5 − 𝑥)3
(a) State the leading term
(b) State the leading coefficient
(c) Sate the degree of the polynomial
(d) What is the end behavior of 𝑃(𝑥)?
(e) What are the different factors of 𝑃(𝑥)?
(f) List all zeros of 𝑃(𝑥) and their multiplicities.
3) Complete the following polynomial identities
(a) 𝑥 2 − 𝑦 2 = ________________________
(b) 𝑥 3 − 𝑦 3 = ________________________
(c) 𝑥 3 + 𝑦 3 = ________________________
(d) 𝑥 4 − 𝑦 4 = ________________________
4) What are the three forms of quadratic function?
Part 2 - Understanding
5) Anna was absent fort the lesson on graphing polynomial functions and asked you for help.
How would you explain to Anna:
(a) How to determine the end behavior of a polynomial function? Why does it work?
(b) What is the correlation between zeros and factors of a polynomial function?
(c) What is the behavior of a polynomial function at x – intercepts?
6) What are the advantages and disadvantages of each form of quadratics function?
7) Given a polynomial equation with real coefficients, how can you determine the number of
complex roots?
Part 3
8) Write equations of the following quadratic functions in all three forms if possible (If not
possible, explain why). Give coordinates of the vertex, state if there is a minimum or
maximum, State the intervals of increase and decrease, find the equation of the line of
symmetry, state the range and zeros of each function.
(b) 𝑦 = 𝑥 2 + 𝑥 + 1
(a)
20
15
10
(0,10)
5
(-5,0)
(1,0)
-10
10
-5
-10
y=f(x)
-15
(c)
X
2
3
4
5
6
Y
0
-0.75
-1
-0.75
0
9) Compare and contrast the quadratic functions from the previous question.
10) Given the graph determine the polynomial function.
11) Construct a rough graph of the function defined by the polynomial
P( x)  x 3  3x 2  4 x  12
12) Find all zeros of P(x) = 2x4 + 9x3 + 3x2 + 36x – 20
13) Detailed mathematical models of the interior of the sun are based on astronomical
observations and our knowledge of the physics of stars. These models allow us to explore
many aspects of how sun “works” that are permanently hidden from the view.
The Standard Model of the sun allows us to investigate many separate properties. One of
these is the density of the heated gas throughout the interior. The function below gives the
best-fit formula, D(x) for the density (in grams/cm3) from the core (x =0) to the surface
(x = 1) and points in-between.
𝐷(𝑥) = 519𝑥 4 − 1630𝑥 3 + 1844𝑥 2 − 889𝑥 + 155
For example, at a radius 30% of the way to the surface, x = 0.3 and so D(0.3) = 14.5
grams/cm3 (source http://spacemath.gsfc.nasa.gov/)
(a) What is the domain of D(x) in this problem?
(b) What is the density of the sun at the surface and what is the density of the sun at its
core?
(c) Is the function increasing or decreasing over the domain?
(d) Use the graphing calculator to estimate the density of the sun at 50% of its radius.