91Ó°ÊÓ

The Factor Theorem is a valuable tool in algebra that connects factors and roots of a polynomial. It states that for a polynomial \(P(x)\), if \((x - c)\) is a factor, then \(P(c) = 0\). Put simply, if plugging a value into the polynomial results in zero, then that value is a root, and \((x - c)\) is a factor.

This theorem simplifies the process of factorization:

• First, find potential roots of the polynomial. These potential roots are often derived from setting the potential factor equal to zero, and solving for \(x\).
• Next, substitute these potential roots into the polynomial.
• If the result is zero, it confirms that the value is a root and the corresponding expression is a factor of the polynomial.

With this understanding, step 2 in the solution attempts to apply this by inserting \(x = 0.5\) into the polynomial. Because the calculation did not yield zero, \((2x - 1)\) is not a factor of the polynomial.

Polynomials are algebraic expressions that consist of variables and coefficients. They are constructed from terms, which are products of a coefficient and a variable raised to a whole-number power. The general form is \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial.

Some key characteristics of polynomial expressions include:

• Degree: This is the highest exponent of the variable in the polynomial. It dictates the number of roots and the general shape of the graph.
• Leading Coefficient: Found in front of the variable with the highest degree; it influences the directional behavior of the polynomial as \(x\) heads towards infinity or negative infinity.
• Roots or Zeros: Values of \(x\) that make the polynomial equal to zero.
• Constant Term: This is the term without a variable, influencing the y-intercept of the graph.

In the exercise, you are working with a sixth-degree polynomial, reflecting its complex nature. Analyzing the individual terms can provide insights into possible factor combinations and behaviors across its real number domain.

Synthetic division is a streamlined technique for dividing a polynomial by another polynomial of the form \(x - c\). It is generally more efficient than traditional long division, especially when dealing with a large polynomial.

The process involves using the roots from factorization attempts to "divide" the polynomial. Here are the steps used in synthetic division:

• Write down the coefficients of the polynomial you are dividing.
• Use the root (from \(x - c = 0\), \(c\) becomes the root) outside the synthetic division structure.
• Bring down the first coefficient, then multiply it by the root and add this product to the next coefficient.
• Repeat until all coefficients are used up.
• If the last number (remainder) is zero, it confirms that \(x - c\) is a factor.

Synthetic division provides a quick method to confirm factors and simplify large polynomial expressions. While not used directly in the step-by-step solution, it is a related and powerful tool for understanding polynomial division techniques.