Question

THOUGHT PROVOKING Write and graph a polynomial function of degree 5 that has all positive or negative real zeros. Label each \(x\)-intercept. Then write the function in standard form.

Short answer

A polynomial function of degree 5 with either all positive or all negative real zeros has been derived. The factored form for negative zeros is \(f(x) = (x+1)(x+2)(x+3)(x+4)(x+5)\), and the standard form is \(f(x) = x^5 + 15x^4 + 85x^3 + 225x^2 + 274x + 120\). The factored form for positive zeros is \(f(x) = (x-1)(x-2)(x-3)(x-4)(x-5)\), and standard form is \(f(x) = x^5 - 15x^4 + 85x^3 - 225x^2 + 274x - 120\).

Step-by-step solution

Select the zeros

Be aware that a polynomial of degree 5 indicates that there are 5 zeros. For simplicity, only use integer zeros. So, the zeros can be \(x = -1, -2, -3, -4, -5\) for all negative solutions, or \(x = 1, 2, 3, 4, 5\) for all positive solutions.

Convert to Factored Form

It's important to transform these zeros into a factored form for the polynomial. If we chose negative zeros, the factored form is \(f(x) = (x+1)(x+2)(x+3)(x+4)(x+5)\). If we chose positive zeros, it would be \(f(x) = (x-1)(x-2)(x-3)(x-4)(x-5)\)

Convert to Standard Form

Expand the factored form of the polynomial to obtain the polynomial in standard form. For negative zeros, this would become \(f(x) = x^5 + 15x^4 + 85x^3 + 225x^2 + 274x + 120\). All coefficients are positive. For positive zeros, this would become \(f(x) = x^5 - 15x^4 + 85x^3 - 225x^2 + 274x - 120\). In this case, the signs alternate.

Graph and Label the Intercepts

Next, graph the polynomial and label each x-intercept, which are the zeros of the polynomial. For the ones we've chosen, the negative zeros will be at points (-1, 0), (-2, 0), (-3, 0), (-4, 0), and (-5, 0). The positive zeros will be at points (1, 0), (2,0), (3, 0), (4, 0), and (5, 0).

Key concepts

Graphing Polynomial Functions

Understanding how to graph polynomial functions is crucial for visualizing the behavior of these equations. A polynomial function of degree 5 will have the general shape of a wave with various peaks and valleys. The graph ultimately is determined by its zeros—the points where the function crosses the x-axis—and the sign of the leading coefficient.

When graphing, take note of the end behavior. For a degree 5 polynomial with a positive leading coefficient, the end behavior shows that the graph will fall to the left and rise to the right. Conversely, if the leading coefficient is negative, the graph will rise to the left and fall to the right. To accurately graph a polynomial function, plot the real zeros, determine the multiplicity of each zero to understand the curve's behavior at these points, and sketch the graph ensuring it follows the end behavior.

Factored Form of a Polynomial

The factored form of a polynomial is an expression where the polynomial is written as the product of its factors, often in terms of \(x - c\), where \(c\) represents each zero of the polynomial. The factored form is incredibly useful as it makes identifying the zeros of the polynomial straightforward. For example, a polynomial function of degree 5 with all negative zeros will look something like \(f(x) = (x + 1)(x + 2)(x + 3)(x + 4)(x + 5)\).

Each factor corresponds to a zero of the polynomial, and setting each factor equal to zero solves for the x-intercepts of the graph. The factored form can also provide insight into the multiplicity of the zeros, though in our case, each zero is distinct, indicating each is a simple zero.

Standard Form of a Polynomial

The standard form of a polynomial is written as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) with decreasing powers of \( x \), where \( a_n \) is the leading coefficient and \( a_0 \) is the constant term. The standard form is helpful for identifying the degree of the polynomial and the leading coefficient, which influence the graph's end behavior and steepness respectively.

To convert a polynomial from factored form to standard form, expand the product of the factors. For example, given a polynomial in factored form with negative zeros \(f(x) = (x + 1)(x + 2)(x + 3)(x + 4)(x + 5)\), multiplying out these factors will result in the standard form, which shows the polynomial as a sum of terms with coefficients.

Real Zeros of a Polynomial

The real zeros of a polynomial are the x-values where the polynomial function crosses or touches the x-axis. These are the solutions to the equation \( f(x) = 0 \). In a graph, they correspond to the x-intercepts. For a degree 5 polynomial, we expect to find up to 5 real zeros.

These zeros are crucial as they play a key role in both the factored form and the graphing of the polynomial. If we know the polynomial has a specific number of positive or negative zeros, this information greatly simplifies the process of writing the polynomial in its factored form and subsequently in standard form. For instance, if a polynomial of degree 5 is given to have all positive real zeros, then all the factors will involve \( (x - c) \) where \( c \) is a positive real number.