Examine if 1 and 2 are zeros of the polynomial `p(x)=x^(3)-3x^(2)-x+3`.

Answer

To determine if 1 and 2 are zeros of the polynomial \( p(x) = x^3 - 3x^2 - x + 3 \), we need to evaluate the polynomial at these values and check if the result is zero.

### Step 1: Evaluate \( p(1) \)

Substituting \( x = 1 \) into the polynomial:

\[ p(1) = 1^3 - 3(1^2) - 1 + 3 \]

Calculating each term:

\[ = 1 - 3(1) - 1 + 3 \]
\[ = 1 - 3 - 1 + 3 \]
\[ = 1 - 3 - 1 + 3 = 0 \]

Since \( p(1) = 0 \), we conclude that 1 is a zero of the polynomial.

### Step 2: Evaluate \( p(2) \)

Now, substituting \( x = 2 \) into the polynomial:

\[ p(2) = 2^3 - 3(2^2) - 2 + 3 \]

Calculating each term:

\[ = 8 - 3(4) - 2 + 3 \]
\[ = 8 - 12 - 2 + 3 \]
\[ = 8 - 12 - 2 + 3 = -3 \]

Since \( p(2) = -3 \), we conclude that 2 is not a zero of the polynomial.

### Conclusion

- \( p(1) = 0 \) → 1 is a zero of the polynomial.
- \( p(2) = -3 \) → 2 is not a zero of the polynomial.

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