Formulas
The sum of the roots of a polynomial p in the form of anxn+an−1xn−1+…+a0 is given by the formula an−an−1.
The product of the roots of p in the form anxnan−1xn−1a0 is (−1)nana0.
Here are some practice problems.
Practice Problem
(2002 AMC 10B Q10)
Suppose that a and b are nonzero real numbers, and that the equation x2+ax+b=0 has solutions a and b. Then the pair (a,b) is ...
Here's a harder problem.
Practice Problem
(2001 AIME 1 Q3) Find the sum of the roots, real and non-real, of the equation
x2001+(21−x)2001=0,
given that there are no multiple roots.
Here's two more examples.
Practice Problem
(2000 AMC 10 Q24)
Let f be a function for which f(3x)=x2+x+1. Find the sum of all values of z for which f(3z)=7.
Practice Problem
(2003 AIME 2 Q9)
Consider the polynomials P(x)=x6−x5−x3−x2−x and Q(x)=x4−x3−x2+1. Given that z1,z2,z3, and z4 are the roots of Q(x)=0, compute P(z1)+P(z2)+P(z3)+P(z4).
Final Notes and Tips
- It's pretty rare to get a question just on Vieta's, usually it's a combination of other stuff (Ex. Binomial Theorem from the 2nd example) unless you're writing a contest for lower grades.
- You should memorize this formula.