Roots of polynomials of degree 3 - cubics
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Roots and linear factors
First we establish that any polynomial with distinct roots can be written as a product of linear factors. Theorem [Equation goes here - download the original to see it.] This theorem can be proven by mathematical induction. Repeated roots The function [Diagram goes here - download the original to see it.] [Diagram goes here - download the original to see it.] So we can see that [Equation goes here - download the original to see it.] at the root . To prove this generally, let [Equation goes here - download the original to see it.] be a function with a repeated root, then [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] So the criterion to test whether a function has a repeated root at is [Equation goes here - download the original to see it.]. The factor theorem can be used to find the roots of a polynomial function and then [Equation goes here - download the original to see it.] is a repeated root can be used to test for repeated roots if necessary. Properties of roots of polynomial equations You are reminded of the properties of roots of quadratic equations, which are dealt with in a separate unit.
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Contents of
Roots of polynomials of degree 3 - cubics
1 Roots and linear factors
2 Cubic equations
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