SOLUTION: Given x^3 - 4x^2 + 2x + 1 =0. b. How many possible negative roots are there? c. What are the possible rational roots? d. Using synthetic substitution, which of the possible ra

Question 176377: Given x^3 - 4x^2 + 2x + 1 =0.
b. How many possible negative roots are there?
c. What are the possible rational roots?
d. Using synthetic substitution, which of the possible rational toots is actually a root of the equation?
e. Find the irrational roots of the equation. (Hint: Use the quadratic formula to the the depressed equation.)
Answer by Mathtut(3670) (Show Source): You can put this solution on YOUR website!
If the coefficients of a polynomial are integers, it is natural to look for roots which are also integers. Any such root must divide the constant term.
:
Given
:
our constant term is 1 ...so 1 and -1 are the only possibles
:
lets try 1:so 1 is a factor
:
lets try -1:so -1 is not a factor
:
This relationship is always true: If a polynomial has rational roots, then those roots will be fractions of the form (plus-or-minus) (factor of the constant term) / (factor of the leading coefficient). However, not all fractions of this form are necessarily zeroes of the polynomial. Indeed, it may happen that none of the fractions so formed is actually a zero of the polynomial.

since 1 is our constant and 1 is our leading coefficient the only possible rational roots are 1 divided by 1 or -1 which is the same as those terms we have already tested.
:

 
   |
  1|1 -4  2   1
   |___1_-3__-1___________________
    1 -3 -1   0
:

so when we factor out the root 1 we get
:

:and using the quadratic formula we get 2 irrational roots from this depressed equation
and

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=13 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 3.30277563773199, -0.302775637731995. Here's your graph:

RELATED QUESTIONS
Given x^3 - 4x^2 + 2x + 1 =0 (a)How many possible positive roots are there? (b)How... (answered by stanbon)

Hi, I need some help with my American School Algebra 2 work. Given X^*3-4x^*2+2x+1=0 A. (answered by Alan3354)

given x^3 - 4x^2 + 2x + 1= 0 (a)how many possible positive roots are there? (b)what are... (answered by solver91311)

Given x^3-4x^2+2x+1=0 A. How many possible negative roots are there? B. Using... (answered by Mathtut)

Please help me. I'm stumped on 2 problems in Algebra 2. Here they are: 1. Find the... (answered by stanbon)

given x^3-4x^2+2x+1=0 how many POSSIBLE POSITIVE AND NEGATIVE ROOTS ARE THERE. HOW MANY... (answered by richwmiller)

Please help. Let p(x)=x^3+7x^2+7x-15 1. How many roots are there? 2. How many... (answered by ikleyn)

what are the possible rational roots of x^3+4x^2+2x+1=o how do you do... (answered by stanbon)

Given the equation x^3 - 2x^2 + x - 3 = 0, what are the possible rational roots please... (answered by jim_thompson5910)