Problem 60 - Algebra and Trigonometry

Question

\(59-62\) . Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros \(-2,0,2,4\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = x^4 - 4x^3 - 4x^2 + 16x \).

Step 1: List the Zeros

The problem provides the zeros of the polynomial: \(-2, 0, 2,\) and \(4\). These values indicate where the polynomial will equal zero.

Step 2: Write the Factors

For each zero \(z_i\), the corresponding factor of the polynomial is \((x - z_i)\). For the provided zeros, the factors are: - \((x + 2)\) for \(-2\)- \(x\) for \(0\)- \((x - 2)\) for \(2\)- \((x - 4)\) for \(4\)

Step 3: Form the Polynomial

Multiply the factors to form the polynomial: \[ f(x) = (x + 2)(x)(x - 2)(x - 4) \] This represents the polynomial with the given zeros and of the specified degree, \(4\).

Step 4: Multiply the Factors

First, multiply the pairs:\[ (x + 2)(x - 2) = x^2 - 4 \] (difference of squares)Next, multiply the remaining:\[ (x)(x - 4) = x^2 - 4x \] Then, combine:\[ f(x) = (x^2 - 4)(x^2 - 4x) \]

Step 5: Expand the Polynomial

Distribute to expand \((x^2 - 4)(x^2 - 4x)\):\[ x^2(x^2) + x^2(-4x) - 4(x^2) - 4(-4x) = x^4 - 4x^3 - 4x^2 + 16x \] Thus the polynomial is:\[ f(x) = x^4 - 4x^3 - 4x^2 + 16x \]

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial are quite important. They are the values of the variable that make the polynomial equal to zero. For example, in our exercise, we have the zeros

• -2
• 0
• 2
• 4

These tell us where the polynomial will "hit" the x-axis. Zeros are also sometimes called roots or solutions. Zeros indicate important interactions on a graph, such as where the graph crosses or touches the x-axis.
To find these zeros, usually you are given a polynomial function and set it equal to zero. However, in this exercise, the zeros are already given, which simplifies your task. Your job is to use these zeros to construct the polynomial.

Degree of a Polynomial

The degree of a polynomial is another essential concept. It refers to the highest power of the variable in the polynomial. In this case, our polynomial’s degree is 4. This means the highest power of our variable, x, in our function is four.
The degree is important because it predicts many aspects of the polynomial's graph:

• The number of possible roots or zeros.
• The number of times the graph might cross or touch the x-axis.
• The general shape or turning points of the graph.

In practice, for this problem, since we are asked to find a polynomial of degree 4 with four given zeros, our task is to create a polynomial that incorporates these features by including a product of four linear factors.

Polynomial Expansion

Polynomial expansion involves multiplying out the factors to write the polynomial as a sum of terms. When given zeros and a degree, such as in our case, you start with factors derived from those zeros:

• For each zero, , the corresponding factor to use is <(x - )>.
• The zeros -2, 0, 2, and 4 give the factors <(x + 2)>, , <(x - 2)>, <(x - 4)>.

Once the factors are established, you multiply them. Polynomial expansion involves systematically distributing the terms to eliminate parentheses and obtain the polynomial's standard form.
For example, from our solution, expanding the factors \[(x + 2)(x)(x - 2)(x - 4)\] results in \[x^4 - 4x^3 - 4x^2 + 16x\]. This expanded form is then used to evaluate or graph the polynomial.