Question

Could someone help me please? I am completely lost on how to figure out this problem.

Use Synthetic division to show that x is a solution for the third degree polynomial equation and use the result to Factor the Polynomial Completely. List All real solutions of the equation

x^3 - 57x - 56 = 0,    x = -1

x = _______ (smallest value)
x = _______
x = _______ (largest value)

Answer 1

x^3 - 57x - 56=0⇒ x(x^2-57)= 56

Verify if x= - 1 is a solution of this equation.
(-1)* [(-1)^2 - 57]=56
(-1)[1 - 57] = 56
(-1)*(-56) = 56

56 =2*28 = (-2)*(-28) = 4*14 = (- 4)*(-14) = 8*7= (-8)*(-7)
x=8 ⇒ 8*(64-57)=56 ok⇒ x=8 is a solution
x= -7 ⇒ (-7)*(49-57)=56 ok ⇒ x= - 7 is a solution

the smallest x= -7
x= - 1
the largest x= 8

Answer 2

From the factor theorem that says, "The polynomial P(x) has x-r as a factor if and only if r is a root of the equation P(x) = 0." So if you plugged -1 into the x's in the equation and get 0 back then, x+1 is a factor. Let's do it.

`( -1)^(3) -57(-1)-56=-1+57-56=0`
It works. When you plugged -1 into x, you got 0 back and that's what the factor theorem says.
Now let's test it with synthetic division. If we divide x+1 into the original equation and get a remainder of 0, then we should be good. I'll write it up on Paint and upload it.
As you can see, our remainder is 0 and our quotient is
`x^(2) -x-56`. This is the Factor the Polynomial part that you wanted. We can factor this into (x-8)(x+7). So our 3 roots of the equation are (x+1)(x-8)(x+7)
Hope this helps and i'm so sorry for the long reply. I forgot to put 0x^2 when I was doing synthetic division. :/ and also sorry for the unclear paint image. you can zoom in if you want. it's nothing really important it just shows me doing the synthetic division part where i divide x+1 into the original equation.