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1. Find the zeros of the polynomial function \[ f(x)=x^{4}-2 x^{3}+4 x^{2}-8 x ...

May 20, 2024

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Solution by Steps

step 1

To find the zeros of the polynomial function

f(x) = x^4 - 2x^3 + 4x^2 - 8x

, we start by factoring out the common term

x

step 2

Factor out

x

from the polynomial:

x(x^3 - 2x^2 + 4x - 8) = 0

step 3

Set each factor equal to zero:

x = 0

and

x^3 - 2x^2 + 4x - 8 = 0

step 4

Solve the cubic equation

x^3 - 2x^2 + 4x - 8 = 0

. This can be done by finding the roots of the cubic polynomial

step 5

Using the Asksia-LL calculator result, the roots of the cubic polynomial

x^3 - 2x^2 + 4x - 8 = 0

are

x = 2

and

x = \pm 2i

step 6

Combine all the roots:

x = 0

x = 2

, and

x = \pm 2i

Answer

The zeros of the polynomial function

f(x) = x^4 - 2x^3 + 4x^2 - 8x

are

x = 0

x = 2

, and

x = \pm 2i

Key Concept

Finding the zeros of a polynomial function

Explanation

To find the zeros of a polynomial function, factor the polynomial and solve for the values of

x

that make each factor equal to zero.