Possible Rational Zeros? 10 Points for Explanation!?

Question

List all the possible rational zeros of f(x) = 3x^3 - x^2 + 2x + 2.

Here's what I got:
x = -0.8, x = -2.9, and x = -1.5

Can you please show me where I went wrong and how to fix it? Thanks so much!

Answer 1

f(x) = 3x^3 - x^2 + 2x + 2.

List of all rational zeroes are { +1,-1 +1/3, -1/3, + 2/3, -2/3. + 2, - 2}

f(-0.56661) = 3(-0.56661)^3 - (-0.56661)^2 + 2(-0.566661) + 2.

= -0.54572513859834 - 0.32104689321 - 1.13322 + 2

= -1.99999203180834 +2.0

= 0.0000096819166......................

Hence the x-intercept is in 0.5666113232 ...........Ans

Answer 2

The Rational Roots Theorem states that if a polynomial has

-- integer coefficients, and

-- an integer, non-zero constant term

then all of the rational roots will be of the form ±(p/q) where p is an integer factor of the constant and q is an integer factor of the highest-order coefficient.



The polynomial f(x) = 3x³ - x² + 2x + 2 does have integer coefficients and an integer, non-zero constant term, so the theorem applies.



The constant is 2. Its factors are 1 and 2

The highest-order coefficient is 3. Its factors are 1 and 3.



The possible rational roots are ±1/1, ±1/3, ±2/1, and ±2/3.



Reducing to lowest terms and listing in numerical order, the possible rational roots are ±1/3, ±2/3, ±1, and ±2.

Answer 3

First, you want to make certain the indications of the coefficients (so as). they are:       f(x)=x^3+2x^2-13x+10       + + - + Now there are 2 places the place it transformations. this suggests the optimal form of useful zeroes. Now you look at f(-x) and do an identical situation. useful roots of f(-x) would be detrimental roots of f(x).       f(-x)= (-x)^3+2(-x)^2-13(-x)+10       f(-x)= -x^3+2x^2+13x+10       - + + + there is in basic terms one sign substitute right here, so there may well be in basic terms one detrimental root for f(x). In all, there may well be in basic terms 3 roots complete because of the fact that this may well be a level 3 equation. meaning we've desperate that they ought to be precisely one detrimental and a pair of useful (in the event that they are all actual). Now use the rational roots theorem to let us know that the only obtainable ration roots are:       ± 10/a million, ± 5/a million, ± 2/a million , ± a million/a million enable's commence plugging interior the negatives first:       f(-a million) = 24       f(-2) = 36       f(-5) = 0 ok, it incredibly is the only detrimental root, enable's circulate to the positives:       f(a million) = 0       f(2) = 0 effective, we are executed. no could desire to learn the rest. The roots are 2, a million, and -5.

Answer 4

The Rational Root Theorem says that the possible rational zeros are all of the form



(divisors of constant term)/(divisors of leading term)



The divisors of your constant term are -1,-2,1, and 2



the divisors of your leading term are -1,-3,1, and 3



So the possibilities are 1,-1, 2,-2, 1/3,-1/3, 2/3, and -2/3. You would usually just write this as (plus or minus) 1,2,1/3, and 2/3.



Note that none of these is actually a zero This polynomial has one irrational zero and two imaginary zeros. The rational root theorem just gives you the possibilities which might work.

Answer 5

factors from 2 are ±1, ±2

factors from 3 are ±1, ±3

Possible rational zeros are factors from 2 divide by factors from 3

so which are ±1, ±2, ±1/3, and ±2/3

To find which one is the zero, you have to see if f(x) = 0 for each of those.

The faster way is by using synthetic divide.