91Ó°ÊÓ

When we talk about the zeros of a polynomial, we are referring to the values of the variable (usually \( x \)) that make the polynomial equal to zero.
For example, in the case of the polynomial given in the original exercise, the zeros are \( 0, 1, 3, 5, \) and \( 10 \). These zeros inform us about where the graph of the polynomial touches or crosses the x-axis. Each zero represents a solution to the equation \( P(x) = 0 \).
If you set each factor of the polynomial to zero, you get the zeros:

• For \( x \): \( x = 0 \)
• For \( x-1 \): \( x = 1 \)
• For \( x-3 \): \( x = 3 \)
• For \( x-5 \): \( x = 5 \)
• For \( x-10 \): \( x = 10 \)

Thus, finding the zeros of a polynomial is like solving a puzzle, where each piece leads you to understanding the factors of the polynomial.

The degree of a polynomial is one of its most important characteristics. It tells us the highest power of \( x \) in the polynomial. The degree also helps us understand the polynomial's general behavior and shape. In the original exercise, the polynomial is represented by its factors: \( x(x-1)(x-3)(x-5)(x-10) \). To determine its degree, we count the number of factors. Each factor \((x - a)\) contributes a power of 1 to the degree.
This means:

• Factor \( x \) contributes 1.
• Factor \( x-1 \) contributes 1.
• Factor \( x-3 \) contributes 1.
• Factor \( x-5 \) contributes 1.
• Factor \( x-10 \) contributes 1.

Adding these contributions together, we find the degree of the polynomial is 5.
This degree is minimal because it exactly matches the number of distinct zeros provided.

Factoring polynomials involves expressing a polynomial as a product of its factors. Each factor is typically of lower degree than the original polynomial.
In the process of solving polynomial equations, factoring helps us identify the zeros.In the provided exercise, the polynomial \( P(x) = x(x-1)(x-3)(x-5)(x-10) \) is already presented in its factored form.
This form swiftly shows us the polynomial's zeros. Each factor contributes a zero to the equation:

• \( x = 0 \)
• \( x = 1 \)
• \( x = 3 \)
• \( x = 5 \)
• \( x = 10 \)

Factoring is a valuable skill as it simplifies the polynomial and elucidates the polynomial's structure. Knowing how to factor effectively allows students to solve polynomial equations efficiently and understand their solutions better.