2.6 theorems about roots of polynomial equations
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Sec 3 Honors 2.6 filled in notes.notebook
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2.6 Theorems About Roots of Polynomial Equations
• Find all possible and actual roots of polynomials• Use polynomial identities to write expressions in different
forms
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Remember the difference between Rational and Irrational?
A rational number is one that can be written as a fraction (examples are 3, ½, 1.25, .333…).
An irrational number is one that will have non‐terminating or non‐repeating decimals (an example is pi, or the square root of 3).
It is important to note that this theorem only finds RATIONAL roots, so if we have irrational or imaginary roots, this does not find them.
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There appears to be a zero between 1 & 1/2 on the graph. The only possible rational roots in that interval on our list are 1/3 & 1/6, so let's try them in synthetic division:
Since neither one gives us a remainder of zero, they are not roots of the function. Therefore, the function does not have any rational roots. It has only irrational & complex roots.
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Steps to find the roots:• List all of the possible rational roots (remember that they will have the form:
factor of constant term / factor of the leading coefficient (p/q's).
• Graph the function on your calculator, and look at the zeros. Do not find the zeros, just look at which possible rational roots in your list could be the right ones!
• Show/prove the rational roots using synthetic division. Continue finding roots and dividing until you have a quadratic (second degree polynomial), then you can factor or use the Quadratic formula to find any remaining roots.
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You can see on the graph that this function has 3 zeros & one looks like it is at x=2, so let's to synthetic division with 2 & see the result.
Sec 3 Honors 2.6 filled in notes.notebook
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After looking at the graph on the calculator, it looks like 1 might be a root, so do synthetic division with it.
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Think about it! When we were solving polynomials in Section 2.4, did we ever get an irrational or complex root that didn’t have it's opposite as an answer, too?
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Remember: if you have a function with rational coefficients & you have an irrational or complex root, you must have it's conjugate, too!
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2.6 Lab Using Polynomial Identities
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Assignment 2.6: prob. 11 –20, 2427, 30, 34, & 35
(don’t do #29 or #32) AND 2.6 Lab: prob. 1014 (don’t do #69 and 15)