find all the zeros of the function?

Question

f(x) = 4x^3 - 3x - 1

can someone explain the steps ?
i don't understand how to find the zeros for this type of function.
thanks. (=

Answer 1

p/q = ±1, ±1/2, ±1/4



Try synthetic division with all of these numbers. +1 works, so one of the factors is x-1, and the other is (4x^3 - 3x - 1)/(x-1) = 4x²+4x+1



(x-1)(4x²+4x+1)

(x-1)(2x+1)²

Answer 2

Since it's 3rd degree, you can't use the usual quadratic formula stuff, you need to find a zero first.



But the numbers make it easy... if x = 1, you get 4 - 3 - 1 = 0. So = 1 is a zero, and that also means x - 1 is a factor of 4x^3 - 3x - 1.



So divide it! The long division is hard to type, I'll do my best...



````````_4x^2_+_4x_+_1___

x - 1 ) 4x^3 + 0x^2 - 3x - 1

````````-(4x^3 - 4x^2)

````````````````````4x^2 - 3x - 1

``````````````````-(4x^2 - 4x)

``````````````````````````````1x - 1

``````````````````````````````-(x - 1)

`````````````````````````````````````0



Needed something to line stuff up, that's what the ``` is all about. The forum won't allow leading spaces!



So you get (4x^2 - 3x - 1)/(x-1) = 4x^2 + 4x + 1, and you should be able to factor this one to find the other two zeros!

Answer 3

Since it's a cubic, we need to either use a trick or guess a solution. We can see that x=1 is a zero of the function ( 4*1 -3*1 - 1 = 0)



Since x = 1 is a zero, x-1 = 0, and

f(x) = (x-1) * (stuff)



To find what the "stuff" is, we need to do long division of polynomials.

stuff = (4x^3 - 3x - 1) / ( x - 1) I can't really show you how to do the long division here due to the type format, but I will try to explain it.

_____________

x-1 ) 4x^3 - 3x - 1 Now, x goes into 4x^3 4x^2 times, so our candidate answer is: 4x^2 + ....

-4x^3 +4x^2 Multiply 4x^2 * (x-1) and subtract from the line above.

4x^2 -3x x goes into 4x^2 4x times. Our revised candidate is: 4x^2 +4x + ...

-4x^2 +4x Multiply through again to get this line

x - 1 Subtract to get the x, then bring the -1 down from above

X-1 goes into x-1 exactly 1 time, so our solution is:



f(x) = (x-1) * (4x^2 + 4x + 1)



So f(x) = 0 if (x-1) = 0 (4x^2 + 4x + 1) = 0



The first part gives x=1 as a zero.



The second part factors as (2x+1)^2 = 0

This gives you a repeated root of x = -1/2



In summary, roots: x = 1, x = -1/2



Hope this helps :-)

Answer 4

f(x)=4x^(3)-3x-1



If a polynomial function has integer coefficients, then every rational zero will have the form (p)/(q) where p is a factor of the constant and q is a factor of the leading coefficient.

p=1_q=4



Find every combination of \(p)/(q). These are the possible roots of the polynomial function.

\1,\(1)/(2),\(1)/(4)



Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is 0, which means it is a root.

4(1)^(3)-3(1)-1



Simplify the expression. In this case, the expression is equal to 0 so x=1 is a root of the polynomial.

0



Since 1 is a known root, divide the polynomial by (x-1) to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

(4x^(3)-3x-1)/(x-1)



Complete the synthetic division of ((4x^(3)-3x-1))/((x-1)).

1,4,0,-3,-1:1,1,4,4,1:1,4,4,1,0



Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by 1.

4x^(2)+4x+1



Solve the equation to find any remaining roots.

x=-(1)/(2)



The polynomial can be written as a set of linear factors.

(x-1)(x+(1)/(2))(x+(1)/(2))



These are the roots (i.e. zeros) of the polynomial 4x^(3)-3x-1.

x=1,-(1)/(2),-(1)/(2)



i hope this helped you understand

Answer 5

We know that 1 is a root of f(x).

Divide 4x^3 - 3x - 1 by x - 1.

You will get 4x^2 + 4x + 1.

Factorize.

(2x + 1)(2x + 1). So the other root is – 1/2.

Answer 6

well 1 is a zero, so:

f(x) = (x-1) (4x^2 + 4x + 1)

now you look for the zeros of (4x^2 + 4x + 1)

f(x)= (x-1) ((2x)^2 + 4x + 1)

f(x)= (x-1) (2x + 1)^2

the zeros are: 1 and -0.5

Answer 7

(X-1)(x-1)^2

Answer 8

I have a TI-83 texas instrument calculator, so it may be different if you aint using the same one.



First, you have to enter the equasion into the calc. by pushing the Y= button.



Second, after entering the equasion, you must graph it, then hit (2nd) (Calc), pick zeroes, make sure the arrows are on the left then hit enter, the nmake sure the arrows are on the right, the nhit enter twice,



then you will get our answers.