1. General Results for Polynomial Equations In this section, we will state five general theorems about polynomial equations, some of which we have already talked about. The first theorem, known as the fundamental theorem of algebra, is a cornerstone of much advance work in math.

2. The Fundamental Theorem of Algebra: In the complex number system consisting of all real and imaginary numbers, if P(x) is a polynomial of degree n, then the equation P(x) = 0 has exactly n roots (provided a double root is counted as 2 roots, a triple root is counted as 3 roots, and so on).

3. Complex Conjugates Theorem: If P(x) is a polynomial with real coefficients, and a + bi is an imaginary root of the equation P(x) = 0, then a – bi is also a root.

5. Theorem 4: If P(x) is a polynomial of odd degree with real coefficients, then the equation P(x) = 0 has at least one real root. If we apply the fundamental theorem of algebra and the complex conjugates theorem to a cubic polynomial with real coefficients, we can say that the polynomial has either three real roots or one real root and a pair of imaginary roots. In any case, the polynomial must have at least one real root.

6. Theorem 5: For the equation The sum of the roots is: The product of the roots is:

7. For the equation: find the sum of the roots and the product of the roots sum =product = For the equation: find the sum of the roots and the product of the roots sum =product =

8. Applying theorem 5 to the general quadratic equation: We know that the sum of the roots is The product of the roots is so we can write we see the equation has the form Ex. 1. Find a quadratic equation with roots Note that this will only work for a quadratic.

9. Ex. 2. Find a cubic equation with integral coefficients that has no quadratic term and as one of the roots. If there is no quadratic term the sum of the roots is 0. or multiply the factors

10. First find the quadratic.