Find all rational roots of the equation.

Give me a minute to type out what I already have.

Question

Find all rational roots of the equation.

Give me a minute to type out what I already have.

savy · Answer

$$4x^4-x^3-2x^2-4x-72$$
There will be 4 total roots; Using Descartes rule of signs; there can be 1 positive root, 3 or 1 negative root; and 0 or 2 non-real.

savy · Answer

I used p/q to figure out the possible roots (which is many so I don't want to type them all out); Used synthetic division to find out that -2 and 9/4 are roots which satisfies the 1 pos 1 neg; so how would i find out if it had non-real and what the non-real roots are?

welshfella · Answer

2 factors  are (x + 2)  and (9x - 4)

So divide (x + 2)(9x - 4) into  the equation to get another quadratic  to solve.

faiqraees · Answer

$\color{#0cbb34}{\text{Originally Posted by}}$ @welshfella
2 factors  are (x + 2)  and (9x - 4)

So divide (x + 2)(9x - 4) into  the equation to get another quadratic  to solve.
$\color{#0cbb34}{\text{End of Quote}}$
If 9/4 is a root then shouldn't the factor be (4x-9)
x=9/4
4x=9
4x-9=0

faiqraees · Answer

(x+2)(4x-9)(ax²+bx+c)=provided polynomial
Compare coefficients and you will get a quadratic equation
Use
$$\large\rm x=\frac{-b±\sqrt{b^2-4ac}}{2a} $$
on the quadratic equation to find out the remaining roots

welshfella · Answer

yea  the factor should be (4x - 9)  My mistake.

savy · Answer

-2,-2,\frac{9}{4},2i,-2i I plugged this into my answer for my homework and it says incorrect. (It says to include repeated answers thats why -2 is listed twice)

I also looked in mathway to be sure I did everything correct and it said this was the answer?
@faiqraees @welshfella

savy · Answer

@ganeshie8 @pooja195 @Hero

welshfella · Answer

The 2 roots -2 and 9/4 are correct.
If we divide  (x +2)(4x-9) into the function we get the quotient x^2 + 4  
Solving  x^2 + 4 = 0  gives 2 non-real roots :-

2i and -2i

welshfella · Answer

This is a quartic equation and has 4 roots not 5.  -2 is not repeated.

faiqraees · Answer

Repeated roots are those roots which have a power on them when in factorized form. (Or you can say that the derivatives also has the same factor as its root)

welshfella · Answer

yes if -2  were repeated there would be a factor (x + 2)^2

savy · Answer

Ok, but it said the answer was incorrect and I have no more attempts. So, I'm confused as to what the answer was. I have my final tomorrow...