Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=x^{2}-25\)

Step-by-Step Solution

Answer

The real zeros of the function are \(x = 5\) and \(x = -5\), both have multiplicity 1.

Step 1: Set the Function to Zero

To find the zeros of the function \(f(x) = x^2 - 25\), the function is first set to zero: \(0 = x^2 - 25\)

Step 2: Solve for x

In the second step, the equation \(0 = x^2 - 25\) is solved using basic algebra. This can be achieved by moving -25 to the other side of the equation and finding the square root: \(x = \sqrt{25}=±5\).

Step 3: Determine the Multiplicity

In the third step, the multiplicity of each zero is determined. As there are no repeat roots and both \(x=5\) and \(x=-5\) appear once as a solution, the multiplicity of each is 1.

Key Concepts

Zeros of Polynomials

Finding the zeros of a polynomial function means identifying the values of \(x\) for which the function equals zero. This process involves solving the equation \(f(x) = 0\). For polynomial functions like \(f(x) = x^2 - 25\), this task begins by setting the entire polynomial equal to zero.
During the solution, algebraic methods can be used. For instance, with \(x^2 - 25\), we can recognize it as a difference of squares, expressed as \(x^2 - 5^2\). This allows us to factor the equation into \((x - 5)(x + 5) = 0\).

To complete the process, each factor is set to zero: \(x - 5 = 0\) and \(x + 5 = 0\), resulting in \(x = 5\) and \(x = -5\) as the zeros of the polynomial.

Multiplicity of Roots

Multiplicity refers to the number of times a particular root appears as a solution for the polynomial equation. It's an important concept because it influences the shape of the graph at those zeros.

In the case of \(f(x) = x^2 - 25\), both solutions, \(x = 5\) and \(x = -5\), are distinct with no repetition within the factorization \((x - 5)(x + 5)\). Because each root appears only once, they both have a multiplicity of 1.

• Multiplicity of 1: The graph crosses the x-axis at the real zero.
• Higher multiplicity: The graph "touches" the x-axis but doesn't cross when its an even multiplicity.

Thus, when a root is repeated, such as \((x - 5)^2 = 0\), \(x = 5\) would have a multiplicity of 2, indicating the graph would only touch the x-axis at \(x = 5\). However, in our exercise, each zero appears only once, leading to a straightforward crossing at each found point.

Graphing Utilities

Graphing utilities are powerful tools that students and mathematicians use to visually confirm the results derived from algebraic processes. These tools make it easy to showcase the behavior of functions over a domain.

For the polynomial \(f(x) = x^2 - 25\), using a graphing calculator or software can help visualize that:

• The graph of \(f(x)\) is a parabola opening upwards due to the highest degree term \(x^2\).
• The x-intercepts (or zeros) are clearly seen at \(x = 5\) and \(x = -5\), confirming the algebraic solution.

These utilities allow students to not only verify their work but also to develop a deeper understanding by allowing them to see the immediate impact of changing polynomial coefficients or altering expressions. The visual representation can effectively highlight the behavior around the zeros and along the curve of the function.