Use synthetic division to find the zeros of the function (must show process)

Question

Use synthetic division to find the zeros of the function (must show process)

f(x)=3x^4-10x^2-25

Answer 1

3x⁴-10x²-25=f(x),,, use z=x² and quadradic formula to find x²=5 is a solution for f(x)=0

divide 3x⁴-10x²-25 by x²-5

equals exaxtly 3x²+5

four zeroes are x=+/-√5, and x=+/-[√(5/3)]*i, ,,,, i=√(-1)

f(x)=3*(x-√5)*(x+√5)*(x-i*√(5/3))*(x+i*√(5/3))

Answer 2

You can try a number of things to start this process: graph the function, eliminate rational roots....

What you will find is that there are no rational roots.

You can use synthetic division to show that there is a root between 2 and 3

3 0 -10 0 -25 | 2

6 12 12 24

------------------

3 6 2 12 -1

3 0 -10 0 -25 |3

9 27 51 153

----------------------

3 9 17 51 128

Now you use a method called Horner's Method for irrational roots.

Reduce the roots by 2 (using synthetic division again) so that the root is between 0 and 1

Use the process again to locate the root between .2 and .3 which gives you the second digit.

Repeat the process until you have the desired number of sig figs.

This is laborious...and today is of academic interest only since a graphing calculator will give you the roots virtually instantaneously.