Find all the rational zeros of the function. (Enter your answers as a comma-separated list.)
f(x) = x^3 – 21x – 20

Question

Find all the rational zeros of the function. (Enter your answers as a comma-separated list.)
f(x) = x^3 – 21x – 20

ash2326 · Answer

We have
$$f(x)=x^3-21x-20$$
We need to find the  roots or zeros of f(x)
There is no direct method of finding the roots, first we'll find one root using hit and trial method. Then well find the other two roots. Let's begin:)

ash2326 · Answer

We have
$$f(x)=x^3-21x-20$$
Let's check if x=1 is a root 
$$f(1)=1-21-20=-40$$
We need to find a value of x for which f(x) is zero
Let's check if x=-1 is a root
$$f(-1)=-1+21+20=0$$
Great:D We found one root x=-1
Let's find the other two, it's easy path now

anonymous · Answer

yea i understand now. i was having a diffulct time understand what to enter in the website as an answer. thanks for your help!

ash2326 · Answer

$$f(x)=x^3-21x-20$$
We have x=-1 as a root so x+1 is a factor, we can divide f(x) by x+1 to find a quadratic , whose roots will be the other two roots of f(x). But I won't divide. I'll show you another way
Let's try to creat x+1 in f(x)
$$f(x)=x^3-21x-20$$
We'll replace 
$$x^3 =x^2(x+1)-x^2$$
so we have
$$f(x)=x^2(x+1)-x^2-21x-20$$
We'll now replace
$$-x^2=-x(x+1)+x$$
so we have
$$f(x)=x^2(x+1)-x(x+1)+x-21x-20$$
or
$$f(x)=x^2(x+1)-x(x+1)-20x-20$$
We'll get
$$f(x)=x^2(x+1)-x(x+1)-20(x+1)$$
Let's take out x+1 common
$$f(x)=(x^2-x-20)(x+1)$$
Let's solve 
$$x^2-x-20=0$$
We need to find factors of -20, which adds  up to -1
-5 and 4 satisfy this
-5+4=-1
-5*4=-20
so
$$x^2-5x+4x-20$$
Take x common from the first two terms and 4 common from the last two terms, we get
$$x(x-5)+4(x-5)=0
$$
Take x-5 common

$$(x-5)(x+4)=0$$
so the other two roots are -4 and 5
So the three roots are
x=-4, 1 and 5