Finding All Roots of a Polynomial Function

Graph f(x) = 8x4 - 50x3 + 101x2 +64x - 195How many real zeroes does this function have?

2 real zeroes.

It has 4 zeroes total: 2 real and

2 imaginary

How do we find the exact values of all the zeroes?

Ex. 1 Find all the zeroes of

f(x)=8x4 - 50x3+101x2+64x - 195

Steps1. Graph f(x)2. Find all real roots (y2=0;

2nd Calc Intersect) If decimal, change to fraction:on home screen hit X,Math,Enter,Enter

3. Use synthetic division to “reduce” equation(use 2nd root on answer from 1st root)

4. Use quadratic formula to find roots of final quadratic.

What are the depressed equations?

Depressed equations are the functions rewritten as a product of factors

For example 1:

f(x)=(2x-3)(8x3-38x2+44x+130)

Practice - Find all zeroes

Ex. 2 f(x)=15x4-134x3+359x2-174x-136

Ex. 3 f(x)=27x4-39x3+22x2+34x-4

Ex. 4 f(x)=14x4-23x3+269x2+977x-145

Answers

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

2. x=4/3,-2/5,4+i,4-i

3. x=-2/3,1/9,1+i,1-i

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