91Ó°ÊÓ

The Factor Theorem is a crucial part of understanding polynomials, especially when it comes to finding their roots. It states that if a polynomial \( f(x) \) has a factor \( (x - a) \), then \( f(a) = 0 \). In simple terms, this means that when you substitute \( a \) into the polynomial, the result should be zero. This gives us a powerful tool for finding the roots of a polynomial.

Let's apply this to our polynomial \( x^3 - 3x^2 + 3x - 1 \). We are informed that \( x - 1 \) is a factor. Using the Factor Theorem, we check by substituting \( x = 1 \) into the polynomial:
- \( 1^3 - 3(1)^2 + 3(1) - 1 = 0 \).

The result is zero, confirming that \( (x - 1) \) is indeed a factor. Thus, the root \( x = 1 \) exists for the polynomial.

Synthetic Division is a fast and efficient method of dividing polynomials, commonly used when dividing by a linear factor. It simplifies the division process, eliminating variables and reducing complex calculations.

To use Synthetic Division for \( x^3 - 3x^2 + 3x - 1 \) by \( x - 1 \), we set it up as follows, using the coefficients of the polynomial:
- Write down 1 (from \( x-1 \)) to the left.
- Draw a horizontal and vertical line to separate the workspace.
- Write the coefficients 1, -3, 3, -1 beside the vertical line.
- Perform the calculations:
- Bring down the first coefficient (1).
- Multiply by the number on the left (1), add to the next coefficient (-3 + 1 = -2), repeat for all coefficients.
- The last number gives the remainder, which should be zero if \( x-1 \) is a factor.

After completing these steps, the resulting quotient is \( x^2 - 2x + 1 \), indicating successful division.

Understanding the multiplicity of roots is essential for analyzing polynomial behavior. A root's multiplicity indicates how many times it appears as a solution.

For the polynomial \( x^3 - 3x^2 + 3x - 1 \), we found that it can be expressed fully as \((x-1)^3\). This indicates that the root \( x = 1 \) has a multiplicity of 3.

Multiplicity affects how the graph behaves at the root:
- If a root has odd multiplicity, the graph will cross the x-axis at the root point.
- If even, the graph only touches or bounces off the x-axis.

In our scenario, since the multiplicity is odd (3), the graph flattens as it passes through the x-axis around \( x = 1 \). This makes the turning point at this root very gentle.