Download CP Algebra 2 Polynomials Review, Connecting the Big Ideas

CP Algebra 2 Polynomials Review, Connecting the Big Ideas, Deducing Consequences, and Applying,
Oh My!
1. State the Factor Theorem.
2. Use the Factor Theorem to determine whether 𝑥 + 3 is a factor of 𝑥 3 + 6𝑥 2 + 11𝑥 + 6.
3. Use the Factor Theorem to determine whether 3 is a root of 𝑥 3 − 4𝑥 2 − 3𝑥 + 18.
4. Why is the Factor Theorem true? Here’s a sketch of a proof, which you will complete.
a. The Division Algorithm for Integers tells us that if P and D are positive integers with P>D, then
P=DQ+r, where Q and r are integers and r<P. To get a sense of this, find Q and r for 16/5 and for 20/4.
b. The Division Algorithm for Polynomials can be expressed in the same way, except here P, D, Q, and r
are polynomials, and instead of comparing numbers by “less than,” we are comparing polynomials by
degree. Write this new Division Algorithm for Polynomials.
c. Using the Division Algorithm for Polynomials, deduce the Factor Theorem. (Note that there are two
parts of the theorem because of “if and only if.”)
5. State the Remainder Theorem.
6. Explain why the Remainder Theorem implies the Factor Theorem.
7. Find the remainder when 2𝑥 2 − 19𝑥 + 24 is divided by 𝑥 − 8.
8. Apply the Factor Theorem and Division to find the zeros of 𝑃(𝑥) = 𝑥 3 − 1.
9. Consider the following work shown in Synthetic Division. Write a mathematical statement that shows
the division of two particular polynomials and its result.
10. Students often misremember whether 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 ) or (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 ),
or something else. Explain how you can figure out or verify which is correct by reasoning.
11. Apply your thinking from #10 to solve the equation 8𝑥 3 + 27 = 0
12. Solve: 𝑥 4 − 3𝑥 2 − 10 = 0.
13. Generalize solving equations like #12: Solve 𝑎𝑥 4 + 𝑏𝑥 2 + 𝑐 = 0.
14. State the Rational Root Theorem.
15. Use the Rational Root Theorem, Factor Theorem, and Division to find all roots of
𝑥 3 + 𝑥 2 − 3𝑥 − 3 = 0.
16. Is √8 a rational or an irrational number? Use the Rational Root Theorem to prove your statement.
17. If 2+√3 is a root of a polynomial with integer coefficients, what other number must be a root of P?
Why?
18. Consider two integers a and b where √𝑏 is irrational and a+√𝑏 is a root of some polynomial P(x)
whose coefficients are all integers. What else must be a root of P(x)? Explain how you know. What if the
coefficients are not integers? Must the number you found still be a root?
19. Multiply 5+√6 by its conjugate.
20. Show that the product of a number of the form in #18 (a+√𝑏) and its conjugate is a real number.
21. Write a polynomial with integer coefficients whose roots include 5+√6 and 2+√3.
22. Is there a second polynomial with integer coefficients whose roots include 5+√6 and 2+√3 ? Why or
why not?
23. If -11+4i is the zero of a polynomial with real coefficients, what other number must be a root?
24. Describe the complex roots of any polynomial with real coefficients.
25. Multiply (3+2i) by its complex conjugate.
26. Prove that the product of a complex number and its conjugate is a real number.
27. State the Fundamental Theorem of Algebra.
28. Prove the theorem is true for any cubic polynomial by considering how the polynomial behaves for
large x and negative x with large absolute value.
29. Using each of the theorems, tools, or ideas below, describe a process to find all the zeros of a
polynomial.
Factor Theorem
Remainder Theorem
Polynomial Division
Rational Root Theorem
Conjugate Pairs
Fundamental Theorem of Algebra
30. Describe how you can use your process described above to prove that the Fundamental Theorem of
Algebra implies that an nth degree polynomial has n roots, when we count the roots with multiplicity and
include complex numbers as possible roots.
31. Find all 4 roots of 𝑥 4 = 1.
32. Find the remainder when (𝑥 + 3)5 + (𝑥 + 2)8 + (5𝑥 + 9)2016 is divided by x+2.
33. If P(x) is a linear polynomial and P(6)-P(2)=12, what is P(12)-P(2)?
34. What else did you learn about Polynomials? State and solve your own problem showing something
you learned.
35. Compare and contrast linear, quadratic, cubic and quartic polynomials. In your discussion, make sure
to include roots, the graphs, and factors.