Problem 61 - College Algebra

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(5 .\) Double zero at \(x=1\) , and triple zero at \(x=3 .\) Passes through the point \((2,15) .\)

Step-by-Step Solution

Answer

The equation is \(f(x) = -15(x-1)^2(x-3)^3\).

Step 1: Identify the Form of the Polynomial

The polynomial has degree 5, with a double zero at \(x = 1\) and a triple zero at \(x = 3\). Therefore, the polynomial can be expressed as: \(f(x) = a(x-1)^2(x-3)^3\).

Step 2: Substitute Point Into the Equation

To find the value of the constant \(a\), use the point \((2, 15)\). Substitute \(x = 2\) and \(f(x) = 15\) into the equation: \(15 = a(2-1)^2(2-3)^3\).

Step 3: Simplify and Solve for a

Simplify the expression: \(15 = a(1)^2(-1)^3\). This results in \(15 = -a\). Therefore, \(a = -15\).

Step 4: Write the Complete Polynomial Equation

Substitute \(a = -15\) back into the polynomial equation: \(f(x) = -15(x-1)^2(x-3)^3\).

Key Concepts

Degree of Polynomial

The degree of a polynomial is an important indicator of the polynomial's behavior. It tells us the highest power of the variable in the polynomial's expression. In simple terms, it's the highest exponent of the variable in the polynomial. For example, in a polynomial equation like \(x^5 - 4x^3 + 2x^2 + x - 7\), the degree is 5, which is the largest exponent on any \(x\) in the polynomial.

This particular degree helps determine:

• The number of roots or zeros the polynomial can have (up to the degree, but not less).
• The end behavior of the polynomial graph. A polynomial of degree 5 will typically have its end points pointing in opposite directions.

In the given exercise, the polynomial is of degree 5, which means that it could have up to 5 real roots, though complex ones could also play a role. Understanding the degree is key to predicting many characteristics of polynomial functions.

Roots and Zeros of Polynomials

Roots, or zeros, of polynomials refer to the values of \(x\) for which the polynomial equals zero. They are the solutions to the polynomial equation \(f(x) = 0\). In the context of graphs, these are the x-intercepts where the polynomial crosses or touches the x-axis.

For example:

• A single zero means the graph crosses the x-axis at that point.
• A double zero, like at \(x=1\) in the given exercise, means the graph just touches the x-axis and bounces back, creating a cusp or a "flattening" effect at \(x=1\).
• A triple zero, like at \(x=3\), implies a point of inflection, where the sign of the slope changes but stays at the x-axis a bit longer.

These zeros are vital to understanding not just where a polynomial graph crosses the axis, but how it behaves around those crossings.

Factored Form of Polynomial

The factored form of a polynomial makes it easier to identify the roots or zeros directly. It is a representation of the polynomial equation as a product of its linear factors.

• For example, if a polynomial has a double zero at \(x=1\), part of its factored form would be \((x-1)^2\).
• If there's a triple zero at \(x=3\), it becomes \((x-3)^3\).

In the given exercise, we've identified these factors from the information provided about zeros. The polynomial's factored form becomes \(f(x) = a(x-1)^2(x-3)^3\), where \(a\) is the constant multiplier adjusted to fit a specific point in the graph, such as the point \((2, 15)\). This pattern allows easy manipulation of the polynomial equation to fit any graph needs by adjusting the constant \(a\).

Polynomial Graphing

Graphing is a powerful tool to visualize polynomial functions and understand their behavior. When plotting a polynomial, several key features are noteworthy:

• The degree of the polynomial, which indicates the potential turning points and end behaviors.
• The zeros of the polynomial, which are the x-intercepts where the graph meets the x-axis.
• The multiplicity of zeros, affecting whether the graph crosses or just touches the x-axis at these points.
• The sign of the leading coefficient, which affects the direction of the opening of the graph.

For the exercise examined, considering the roots and their multiplicity helps in sketching an accurate graph. The double zero at \(x=1\) means a bounce-off at that point, and a triple zero at \(x=3\) implies a "saddle" point. The calculated value of \(a\), found to be \(-15\), affects the width and direction of the curve, reflecting this polynomial's end behavior. Such characteristics are crucial in tracing the graph shape effectively, providing a clear picture of how polynomial behaves graphically across its domain.