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To express \( g(x) = x^3 + x^2 - 3x + 1 \) as a product of linear factors, we can start by using the fact that 1 is given as a zero of the polynomial. This means \( x - 1 \) is a factor of the polynomial. Now, we can perform polynomial division of \( g(x) \) by \( x - 1 \) to factor it.

### Steps:

1. **Divide \( g(x) \) by \( x - 1 \)** using synthetic or long division.
2. The result will be a quadratic polynomial.
3. Factor the quadratic polynomial.
4. The final answer will be a product of linear factors.

Let me proceed with synthetic division first.

### Synthetic Division of \( g(x) \) by \( x - 1 \):

The coefficients of \( g(x) \) are \( [1, 1, -3, 1] \), corresponding to the powers of \( x^3, x^2, x, \) and the constant.

Let's calculate it:

\[
\begin{array}{r|rrrr}
1 & 1 & 1 & -3 & 1 \\
  &   & 1 & 2 & -1 \\
\hline
  & 1 & 2 & -1 & 0 \\
\end{array}
\]

This division gives \( x^2 + 2x - 1 \), with no remainder. So:

\[
g(x) = (x - 1)(x^2 + 2x - 1)
\]

Now, we factor the quadratic \( x^2 + 2x - 1 \) using the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-1)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}
\]

Thus, the factors of the quadratic are \( (x + 1 - \sqrt{2})(x + 1 + \sqrt{2}) \).

### Final Factorization:

\[
g(x) = (x - 1)(x + 1 - \sqrt{2})(x + 1 + \sqrt{2})
\]

This is the expression of \( g(x) \) as a product of linear factors.

Would you like further explanation or have any questions?

### Related Questions:

1. How is synthetic division performed in general?
2. How can we verify that 1 is a root of the polynomial \( g(x) \)?
3. What is the quadratic formula, and how is it derived?
4. How do you factor a quadratic when it has irrational roots?
5. How can you check if a factorization is correct?

### Tip:
When performing synthetic division, always verify that the remainder is zero if the divisor is supposed to be a factor.

For the polynomial g(x) = x^3 + x^2 - 3x + 1, express g(x) as a product of linear factors, given that 1 is a zero.

Learn how to express the cubic polynomial g(x) = x^3 + x^2 - 3x + 1 as a product of linear factors using polynomial division and the quadratic formula. This problem involves core algebraic concepts, suitable for students in Grades 10-12.

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