Problem 280 - Precalculus

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(3 .\) Zeros at \(x=-5, \quad x=-2,\) and \(x=1 . y\) -intercept at \((0,6)\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = -\frac{3}{5}(x + 5)(x + 2)(x - 1) \).

Step 1: Understanding Polynomial Structure

A polynomial of degree 3 is represented in the form of \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \]where \(a\) is a constant and \(r_1, r_2, r_3\) are the roots of the polynomial. The zeros of the polynomial are given as \( x = -5, x = -2, \) and \( x = 1 \), making \( r_1 = -5, r_2 = -2, r_3 = 1 \).

Step 2: Formulating the Polynomial Equation

Substitute the zeros into the polynomial structure:\[ f(x) = a(x + 5)(x + 2)(x - 1) \]

Step 3: Finding the Leading Coefficient 'a'

Using the given \(y\)-intercept \((0, 6)\), substitute \(x = 0\) and \(f(x) = 6\) into the equation:\[ 6 = a(0 + 5)(0 + 2)(0 - 1) \]This simplifies to:\[ 6 = a(5)(2)(-1) \]\[ 6 = -10a \]Solve for \(a\):\[ a = -\frac{6}{10} = -\frac{3}{5} \]

Step 4: Writing the Final Polynomial Equation

Substitute \(a\) back into the polynomial equation to get:\[ f(x) = -\frac{3}{5}(x + 5)(x + 2)(x - 1) \]

Key Concepts

Degree of Polynomial

The degree of a polynomial is one of its most defining characteristics. It tells us the highest power of the variable in a polynomial equation. The degree determines the number of solutions or 'roots' the polynomial can have. For example, in a polynomial of degree 3, the highest power of any term is 3. This particular power informs us that the polynomial can have up to 3 roots.
Understanding the degree is also crucial when predicting the shape and behavior of the polynomial graph. A polynomial of degree 3 typically forms what is known as a cubic graph, which can have one or more turning points. The degree also dictates how many times the graph can intersect the x-axis, with a maximum of 3 intersections for a degree 3 polynomial.

Polynomial Roots

Polynomial roots, also known as zeros or solutions, are the values of 'x' which make the polynomial equal to zero. In simpler terms, they are the x-values where the graph of the polynomial crosses or touches the x-axis. Our example mentions zeros at \( x = -5, x = -2, \) and \( x = 1 \). This means that if we plug these x-values into the polynomial equation, the output will be zero.
Identifying roots is fundamental in writing the polynomial equation. Each root is expressed as a factor in the polynomial, for example, a root at \( x = -5 \) corresponds to the factor \((x + 5)\). Collectively, these factors formed from the roots contribute to constructing the polynomial equation as seen in \( f(x) = -\frac{3}{5}(x + 5)(x + 2)(x - 1) \).

Y-Intercept

The y-intercept is the point where the graph of the polynomial crosses the y-axis. This occurs when \(x = 0\). For this polynomial, the y-intercept is given as \((0, 6)\). Simply put, setting \(x = 0\) in the polynomial equation should yield \(6\).
This is particularly useful in determining the leading coefficient or scaling factor \(a\) of the polynomial equation. Substituting \(x = 0\) into the polynomial provides an equation that can be solved for \(a\). In our example, using the y-intercept \((0, 6)\), the calculated value of \(a\) helps complete the polynomial expression.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power in a polynomial equation. It plays a critical role in affecting the width and direction of the polynomial graph. For instance, if the leading coefficient is positive, the graph will end up in the positive direction on both sides; if negative, it ends in the opposite direction.
In our scenario, after considering the y-intercept, we determined the leading coefficient \(a\) as \(-\frac{3}{5}\). This implies the polynomial equation opens downwards. Hence, the shape and vertical stretch of the graph are influenced by this coefficient, providing an important insight into the behavior of the polynomial's graph.