SOLUTION: Hello. I was hoping you could please help me? I need to solve this equation : k^5+4k^4-32k^3=0 Since its highest degree is 5, my teacher said we should get 5 roots, but wh

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hello. I was hoping you could please help me? I need to solve this equation : k^5+4k^4-32k^3=0 Since its highest degree is 5, my teacher said we should get 5 roots, but wh      Log On


   

Question 403541: Hello. I was hoping you could please help me?
I need to solve this equation :
k^5+4k^4-32k^3=0
Since its highest degree is 5, my teacher said we should get 5 roots, but when I tried it, I only got 3 roots, which were: 0, -8, and 4
Can you please show me the correct way to do it? I'm doing it by factoring...
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!

k%5E5%2B4k%5E4-32k%5E3=0+
k%5E3%28k%5E2%2B4k-32%29=0+....your equation will be equal to 0 if k%5E3=0 or %28k%5E2%2B4k-32%29=0+
so,
if k%5E3=0 ...you have three roots, all of them are equal to 0
k1=0
k2=0
k3=0
if %28k%5E2%2B4k-32%29=0+...here is quadratic, and it has two roots
use quadratic formula to find roots
k+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
k+=+%28-4+%2B-+sqrt%28+4%5E2-4%2A1%2A%28-32%29+%29%29%2F%282%2A1%29+
k+=+%28-4+%2B-+sqrt%2816+%2B128+%29%29%2F2+
k+=+%28-4+%2B-+sqrt%28144+%29%29%2F2+
k+=+%28-4+%2B-+12%29%2F2+

k4+=+%28-4+%2B+12%29%2F2+
k4+=+8%2F2+
k4+=+4+

k5+=+%28-4+-+12%29%2F2+
k5+=+%28-16%29%2F2+
k5+=+-8



Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
There's a nice theorem (fairly difficult to prove though) called the fundamental theorem of algebra that says that any n-degree polynomial has n complex roots, including multiple roots. Therefore we expect five roots.

Factoring the expression, this becomes

k%5E3%28k%5E2+%2B+4k+-+32%29+=+0 --> k%5E3+=+0 or k%5E2+%2B+4k+-+32+=+0

The first equation has a triple root of 0. The second equation can be factored as %28x+%2B+8%29%28x+-+4%29+=+0 --> x = -8, x = 4.