Problem 73 - Algebra 2

Question

Construct a polynomial function with the given zeros. $$ x=5,0,2 $$

Step-by-Step Solution

Answer

The polynomial function with the given zeros of 5, 0 and 2 is \(f(x) = x(x - 5)(x - 2)\)

Step 1: Identify the zeros

In this exercise, the given zeros are 5, 0, and 2.

Step 2: Formulate the factors

Each zero corresponds to a factor in the polynomial function. A zero at \(x = a\) corresponds to a factor of \(x - a\). So here, the factors of the polynomial function are \(x - 5\), \(x - 0\) (or simply \(x\)) and \(x - 2\).

Step 3: Construct the polynomial

To find the polynomial function, simply multiply all these factors together. So, the polynomial function is \(f(x) = (x - 5) * x * (x - 2)\). This can be further simplified to \(f(x) = x(x - 5)(x - 2)\).

Key Concepts

Zeros of Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients.
One of the most important concepts in understanding polynomials is the idea of 'zeros.' Essentially, the zeros of a polynomial are the solutions to the equation when the polynomial is set to zero.
They represent the x-values where the graph of the polynomial crosses the x-axis.

• If you have a zero at \( x = a \), it means that \( a \) is a solution to the equation \( f(x) = 0 \).
• These zeros can be real or complex numbers, depending on the polynomial.
• In this exercise, the given zeros are 5, 0, and 2.

Understanding zeros is crucial because they help us in constructing polynomial functions.
When given the zeros of a polynomial, you can directly relate them to factors of the polynomial function.

Factorization

Once you have the zeros of a polynomial, factorization is the next step. Factorization is the process of breaking down a polynomial into the product of its factors.
This process simplifies the task of constructing the polynomial function.

• Each zero \(x = a\) corresponds to a linear factor of \( (x - a) \) in the polynomial.
• For zero at \( x = 0 \), the factor is simply \( x \), because \( x - 0 = x \).
• Therefore, if we take the zeros 5, 0, and 2, the corresponding factors will be \( (x - 5) \), \( x \), and \( (x - 2) \).

This means any polynomial function containing these zeros can be expressed as a product of these factors.
Factorization not only helps in constructing polynomials but also plays a crucial role in solving them.

Constructing Polynomial Equations

Constructing polynomial equations using their zeros and factors is a straightforward process.
Once you have identified the factors, you can multiply them together to get the polynomial equation.

• The original polynomial equation formed by joining the factors \( (x - 5) \), \( x \), and \( (x - 2) \) is \( f(x) = (x - 5)x(x - 2) \).
• You can expand it to simply \( f(x) = x(x - 5)(x - 2) \).
• This leads to a polynomial of degree three, as there are three zeros.

Constructing a polynomial involves organizing its factors into a coherent product that forms the complete equation.
This is an essential skill, as it allows you to create polynomials that satisfy specific conditions and to solve real-world problems through mathematical modeling.