Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function that has the given zeros. (There are many correct answers.) \(4,-3,3,0\)

Step-by-Step Solution

Answer

The polynomial function that has the zeros 4, -3, 3, and 0 is \(f(x) = x(x^3 - 4x^2 - 9x + 36)\).

Step 1: Interpret the Given Zeros

The problem gives four zeros which are \(4, -3, 3, 0\). Each of these zeros corresponds to a factor in the polynomial function.

Step 2: Convert Zeros to Factors

You can find the factors of a polynomial by setting each zero equal to x and then rearranging the equation to equal 0. In this case, the factors corresponding to the given zeros are \(x-4\), \(x+3\), \(x-3\), and \(x\).

Step 3: Form the Polynomial

The polynomial function can be formed by multiplying all the obtained factors. So, based on this the polynomial function would be \(f(x) = (x-4)(x+3)(x-3)x\).

Step 4: Simplify the Polynomial

The polynomial function can be simplified further by multiplying the individual factors. After multiplying, the polynomial is \(f(x) = x(x^3 - 4x^2 - 9x + 36)\).

Key Concepts

Zeros of a Function

The concept of "zeros of a function" is fundamental in understanding polynomial functions. A zero of a function, often referred to as a root, is an input value that yields an output of zero for that function. In simpler terms, if you plug in a zero into the equation of a function, the result should be zero. This means for a polynomial function, the zeros are the values of \(x\) for which \(f(x) = 0\).

Consider a polynomial with zeros \(4, -3, 3,\) and \(0\). Each of these values indicates that the polynomial touches or crosses the x-axis at these points when graphed. This property helps us understand the behavior of polynomial functions and forms the basis for finding the polynomial equation when the zeros are known.

Factoring Polynomials

Factoring polynomials is the process of expressing a polynomial as a product of its factors, rather than a sum. Each factor is a simpler polynomial, ideally of lower degree.

Let's take the given zeros \(4, -3, 3,\) and \(0\). These zeros tell us how to construct the factors of our polynomial. For each zero, say \(z\), a factor is formed as \(x-z\). Thus, for the zeros provided, the corresponding factors are:\

\
• \(x-4\)
\
• \(x+3\)
\
• \(x-3\)
\
• \(x\)
\

When multiplied together, these factors represent the polynomial in question. The ability to recognize and convert zeros into factors is key in constructing polynomial equations from given zeros.

Multiplying Polynomials

Multiplying polynomials involves expanding the expression resulting from two or more polynomial factors. This involves applying the distributive property several times to ensure every term of each factor is multiplied by every term of the other factors.

Given the factors \((x-4)(x+3)(x-3)x\), to form the corresponding polynomial function, each term must interact with every other term from other factors. This paired expansion ensures that the polynomial covers all possible combinations of products.

Here, one must pay careful attention to the order of operations, ensuring that terms are combined correctly. Combining like terms and maintaining proper coefficients through precise arithmetic is crucial while performing this expansion.

Simplifying Polynomials

The final step in crafting a polynomial function from its factors is simplifying the product. This involves combining like terms to write the polynomial expression in its most concise, canonical form.

Once the factors \((x-4)(x+3)(x-3)x\) are multiplied, you'll have a lengthy expression that may contain terms like \(x^3, x^2, x\), and constant terms. Simplifying involves collecting terms with the same powers of \(x\) and summing their coefficients. The expanded polynomial \( f(x) = x(x^3 - 4x^2 - 9x + 36) \) becomes simplified when like terms are consolidated to give a cleaner form.

This simplification not only makes the polynomial easier to read and understand but also reveals more about its behavior and properties in a condensed manner.