Polynomials I
Find a polynomial that is equal to f(x) = (x − 1)(x − 2)(x − 3).
x3 − 6x2 + 11x − 6
The polynomial 3x6 + 7x3 + x2 + 1 = 0 has how many complex solutions?
The polynomial has 6 complex solutions.
If we know a real solution to an even-ordered polynomial, we know it has at least one more.
True.
Does O(10n10) = O(5n10)?
Yes
Find A and B such that 7/6 = A/2 + B/3.
A=1, B=2
Polynomials above this degree do not have general solutions.
Do degree 5+ polynomials have solutions?
The polynomial 19x5 + 100x2 + 8 = 0 is guaranteed to have how many real solutions?
The polynomial is guaranteed to have 1 real solution.
If 3 + sqrt(2) is a solution to a polynomial then is 3 -sqrt(2) also a solution?
Yes
Find O(4n9+3log(n))?
O(n9)
What is the appropriate partial fraction form for (3x-5)/(x-1)2?
A/(x-1) + B/(x-1)2
Construct a polynomial with degree 3 that has only 1 root in R.
f(x) = (x − a)(x2 + b) where a, b ∈ R and b > 0
Approximate f(x) = (x2 + 4x + 1)/(2x2 + 10x + 3) for x = 10100
f(10100) ~ 1/2
Is 4/(x2 - 4) = 1/(x - 1) + 1/(x + 1) true?
No, this is false.
{4/(x2 - 4) = 1/(x - 1) - 1/(x + 1)}
Arrange from smallest to largest: O(n), O(log(n)), O(nn).
O(log(n)) < O(n) < O(nn)
Any functions of the form f(x)=∑∞n=0 anxn.
What is a polynomial?
If 10 + 5i is a solution to a polynomial, what other solution will it have?
10 - 5i
Determine if f(f(x))) for f(x) = (x + 1)2
f(f(x)) = ((x+1)2 + 1)2
Construct a polynomial with degree 5 that has only 1 root in R−Q (irrational numbers) and 1 repeated root n ∈ Z and 2 purely complex roots (i.e. x = ±ib for b ∈ R)
f(x) = (x − π)(x − n)2(x − ib)(x + ib)
Simplify (x - 1)/(x2 - 1).
1/(x + 1)