This document discusses how to find all the roots, or zeroes, of a polynomial function. It explains that there are typically real and imaginary roots. To find them, one should graph the function, find where it crosses the x-axis for real roots, and use synthetic division and the quadratic formula to factor the polynomial into linear and quadratic factors corresponding to its roots. Examples are provided to illustrate the process of finding all roots of different polynomial functions.

1. Finding All Roots of a Polynomial Function

2. Graph f(x) = 8x 4 - 50x 3 + 101x 2 +64x - 195 How many real zeroes does this function have? 2 real zeroes. It has 4 zeroes total: 2 real and 2 imaginary

3. How do we find the exact values of all the zeroes? Ex. 1 Find all the zeroes of f(x)=8x 4 - 50x 3 +101x 2 +64x - 195 Steps Graph f(x) Find all real roots (y2=0; 2nd Calc Intersect) If decimal, change to fraction:on home screen hit X,Math,Enter,Enter Use synthetic division to “reduce” equation(use 2nd root on answer from 1st root) Use quadratic formula to find roots of final quadratic.

5. What are the depressed equations? Depressed equations are the functions rewritten as a product of factors For example 1: f(x)=(2x-3)(8x 3 -38x 2 +44x+130)

6. Practice - Find all zeroes Ex. 2 f(x)=15x 4 -134x 3 +359x 2 -174x-136 Ex. 3 f(x)=27x 4 -39x 3 +22x 2 +34x-4 Ex. 4 f(x)=14x 4 -23x 3 +269x 2 +977x-145

7. Answers x = 3/2,-5/4,3 +2i, 3-2i x=4/3,-2/5,4+i,4-i x=-2/3,1/9,1+i,1-i x=1/7,-5/2,2+5i,2-5i