SOLUTION: Find the zeros of the polynomial function f(x)=x^7−x^5−16x^3+16x and state the multiplicity of each.

Algebra ->  Test -> SOLUTION: Find the zeros of the polynomial function f(x)=x^7−x^5−16x^3+16x and state the multiplicity of each.       Log On


   

Question 317215: Find the zeros of the polynomial function f(x)=x^7−x^5−16x^3+16x and state the multiplicity of each.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
+graph%28+300%2C+300%2C-5%2C5%2C-8%2C8%2C+x%5E7-x%5E5-16x%5E3%2B16x%29+
As you see from the graph, there are 5 real roots.
x=-2,-1,0,1,2
Each root has multiplicity of 1.
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The polynomial found by these zeros is,

Divide the original polynomial by this polynomial to find the quadratic remainder.
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First factor:x%5E2
x%5E2%28x%5E5-5x%5E3%2B4x%29=x%5E7-5x%5E5%2B4x%5E3
Subtract this from the original polynomial to get the remainder,
%28x%5E7-x%5E5-16x%5E3%2B16x%29-%28x%5E7-5x%5E5%2B4x%5E3%29=4x%5E5-20x%5E3%2B16x
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Next factor:4
4%28x%5E5-5x%5E3%2B4x%29=4x%5E5-20x%5E3%2B16x
Subtract this from the remainder
%284x%5E5-20x%5E3%2B16x%29-%284x%5E5-20x%5E3%2B16x%29=0
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So the remainder quadratic is x%5E2%2B4=0 which has complex roots (-2i,2i)
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x%5E7-x%5E5-16x%5E3%2B16x=%28x%2B2%29%28x%2B1%29x%28x-1%29%28x-2%29%28x%5E2%2B4%29