find the real zeros of the following function f(x)=x^3-5x^2+5x-1

Question

I  need to figure this out for Alegbra II.

Answer 1

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

f (x )=  x³- 5x²+ 5x - 1

If p/q is a rational zero, then p is a factor of and q  is a factor of 1.

The possible values of p are   ± 1.

The possible values for q  are ± 1.

So, p /q = ± 1

Make a table for the synthetic division and test possible  zeros.

p /q	1	-5	5	-1
1	1	-6	11	-12
1	1	-4	1	0

Since f (1) = 0,   x  = 1 is a zero. The depressed polynomial is x²- 4x + 1 = 0

Since the depressed polynomial of this zero, x²- 4x + 1, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation x²- 4x + 1 = 0

Roots are

Zeros of function

Real zero of function f (x) = x³- 5x²+ 5x - 1 is at x  = 1.