Questions: Find all zeros of f(x)=x^3+x^2+x+1. Enter the zeros separated by commas.

Transcript text: Find all zeros of $f(x)=x^{3}+x^{2}+x+1$. Enter the zeros separated by commas.

Solution

Solution Steps

Step 1: Identify the polynomial

The given polynomial is $ f(x) = x^3 + x^2 + x + 1 $.

Step 2: Factor the polynomial

To find the zeros, we need to factor the polynomial. Notice that $ f(x) $ can be factored by grouping: \[ f(x) = x^3 + x^2 + x + 1 = (x^3 + x^2) + (x + 1) \] \[ f(x) = x^2(x + 1) + 1(x + 1) \] \[ f(x) = (x^2 + 1)(x + 1) \]

Step 3: Solve for the zeros

Set each factor equal to zero and solve for $ x $:

$ x^2 + 1 = 0 $ \[ x^2 = -1 \] \[ x = \pm i \] (where $ i $ is the imaginary unit)
$ x + 1 = 0 $ \[ x = -1 \]

Final Answer

The zeros of the polynomial $ f(x) = x^3 + x^2 + x + 1 $ are $ -1, i, -i $.

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