Problem 10 - Precalculus : Building Concepts and Connections

Question

For each polynomial function, list the zeros of the polynomial and state the multiplicity of each zero. $$h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^{7}$$

Step-by-Step Solution

Answer

The zeros of the polynomial \(h(x)\) are \(\sqrt{2}\) and \(-\sqrt{2}\). The multiplicity of the zero \(\sqrt{2}\) is 13, and the multiplicity of the zero \(-\sqrt{2}\) is 7.

Step 1: Identify the Zeros

The zeros of the polynomial are the solutions of the equation \(h(x) = 0\). Looking at the polynomial, we have \(h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^{7}\). If we set each factor equal to zero, we get zero at \(x=\sqrt{2}\) and at \(x=-\sqrt{2}\)

Step 2: Identify the Multiplicity

The multiplicity of a zero is given by the exponent on its term. In \(h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^{7}\), the zero \(\sqrt{2}\) as a multiplicity of 13, and the zero \(-\sqrt{2}\) has a multiplicity of 7

Key Concepts

Zeros of Polynomial

The zeros of a polynomial are the values of the variable which make the polynomial equal to zero. In other words, zeros are the solutions to the equation that is formed when you set the polynomial equal to zero. Finding these zeros is essential, as they tell us where the graph of the polynomial intersects the x-axis. Consider the polynomial given in the example: \[h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^7\]To find the zeros, we set each factor equal to zero:

• First factor: \[(x-\sqrt{2}) = 0\]Solving gives us \[x = \sqrt{2}\]
• Second factor: \[(x+\sqrt{2}) = 0\]Solving gives us \[x = -\sqrt{2}\]

These are the zeros of the polynomial, meaning the function will be equal to zero at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \). Knowing the zeros helps in sketching the graph and understanding the behavior of the polynomial function.

Multiplicity

Multiplicity refers to the number of times a particular zero appears in the polynomial. In mathematical terms, it is the exponent of the factor that corresponds to the zero.In our polynomial \[h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^7\], we observe:

• The zero \( x = \sqrt{2} \) is associated with the factor \((x-\sqrt{2})^{13}\). Here, 13 is the exponent, so the multiplicity is 13.
• The zero \( x = -\sqrt{2} \) is associated with the factor \((x+\sqrt{2})^7\). Here, 7 is the exponent, so the multiplicity is 7.

Multiplicity is crucial because it affects the shape of the polynomial's graph at each zero. A zero with a multiplicity of 1 means the graph crosses the x-axis at that zero, whereas with a multiplicity of 2 or more, the graph will "bounce off" the x-axis at the zero. This behavior makes predicting and understanding the graph much easier.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of its simplest polynomial components, or factors. It's akin to finding the building blocks of your polynomial, showing which numbers multiply together to give you the entire expression.Consider the polynomial we are dealing with:\[h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^7\]This polynomial is already factored into its component binomials \((x-\sqrt{2})\) and \((x+\sqrt{2})\) raised to the respective powers. Factoring offers us several insights:

• It reveals the zeros of the polynomial.
• Shows us the multiplicity as observed in the exponents.
• Helps in taking apart complex formulas to simplify calculation and analysis.

Factoring is a fundamental step in understanding polynomial equations and is often used in solving, simplifying, and drawing the graph accurately. It allows one to see exactly where the polynomial will touch or cross the x-axis, providing necessary information for graphing and analyzing polynomial behavior.