SOLUTION: Solve the following polynomial equation x^5-13x^3+36x=0 One of the following is the correct answer. Which one? A) x=0, x=+or-2, x=+or-3 B) x=1, x=2, x=18 C) x=+or-2, x

Algebra ->  Equations -> SOLUTION: Solve the following polynomial equation x^5-13x^3+36x=0 One of the following is the correct answer. Which one? A) x=0, x=+or-2, x=+or-3 B) x=1, x=2, x=18 C) x=+or-2, x      Log On


   

Question 1198584: Solve the following polynomial equation x^5-13x^3+36x=0
One of the following is the correct answer. Which one?
A) x=0, x=+or-2, x=+or-3
B) x=1, x=2, x=18
C) x=+or-2, x+or-3
D) x=+or-3, x=+or-5
E) x=0, x=+or-3, x=+or-12
Found 3 solutions by MathLover1, MathTherapy, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E5-13x%5E3%2B36x=0..........factor
x+%28x%5E4+-+13x%5E2+%2B+36%29+=+0

factor x%5E4+-+13x%5E2+%2B+36:
let x%5E2=u

Substitute back x%5E2=u
%28x%5E2+-9%29%28x%5E2-+4%29=%28x+-+3%29+%28x+-+2%29+%28x+%2B+2%29+%28x+%2B+3%29
so,
x%5E5-13x%5E3%2B36x=x%28x+-+3%29+%28x+-+2%29+%28x+%2B+2%29+%28x+%2B+3%29+
solutions:
x=0
x=3
x=2
x=-2
x=-3

answer:
A)

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the following polynomial equation x^5-13x^3+36x=0
One of the following is the correct answer. Which one?
A) x=0, x=+or-2, x=+or-3
B) x=1, x=2, x=18
C) x=+or-2, x+or-3
D) x=+or-3, x=+or-5
E) x=0, x=+or-3, x=+or-12
Factor out GCF, x to get: 
               x[x2(x2 - 9) - 4(x2 - 9)] = 0 
                         x(x2 - 4)(x2 - 9) = 0
          x(x - 2)(x + 2)(x - 3)(x + 3) = 0 
                                                <===== CHOICE A)

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

First let's pull out the GCF x
x^5-13x^3+36x = 0
x(x^4-13x^2+36) = 0

Now let's factor x^4-13x^2+36
To do so, we have at least two methods

Method 1)
Let w = x^2
So w^2 = x^4

Then x^4-13x^2+36 is the same as w^2-13w+36
Through trial and error, that would factor to (w-4)(w-9)
Note: -4 and -9 add to -13 and multiply to 36.

Then,
(w-4)(w-9) = (x^2-4)(x^2-9)
(w-4)(w-9) = (x-2)(x+2)(x-3)(x+3)
after applying the difference of squares rule.

So overall,
x^5-13x^3+36x = 0
x(x^4-13x^2+36) = 0
x(x-2)(x+2)(x-3)(x+3) = 0
To find the roots, set each factor equal to zero and solve for x.
Example: x-2 = 0 leads to x = 2 as one root.

Therefore, the five roots are:
x+=+0, x+=+%22%22+%2B-+2, x+=+%22%22+%2B-+3
which is the shorthand way of saying
x = 0, x = -2, x = 2, x = -3, x = 3

Answer: Choice A

--------------------------------------------------------

Method 2)

Here's another way to factor x^4-13x^2+36

Use the rational root theorem to list out the factors of the last term 36:
1, 2, 3, 4, 6, 9, 12, 18, 36
List the negative factors as well
-1, -2, -3, -4, -6, -9, -12, -18, -36

Then through trial and error, you should find that x = -2, x = 2, x = -3, x = 3 are actual roots since they cause x^4-13x^2+36 to be zero.
This gives the factorization (x-2)(x+2)(x-3)(x+3)

I don't recommend this second method as the first method is more efficient. But it's good to keep an alternative in mind.