Problem 29 - Algebra 2

Question

Find the zeros of each function. State the multiplicity of multiple zeros. $$ y=(x+3)^{3} $$

Step-by-Step Solution

Answer

The function has one zero at x = -3 with a multiplicity of 3.

Step 1: Identify the Zero of the Function

The zeros of the function are the values of x for which the function equals zero. To find the zeros of the given function, set the function equal to zero: $$ (x+3)^3 = 0 $$.

Step 2: Solve the Equation

To find the value of x that makes the expression zero, we take the cube root of both sides of the equation. This gives us: $$ x+3 = 0 $$Solving for x gives: $$ x = -3 $$.

Step 3: Determine the Multiplicity

The multiplicity of a zero is determined by the exponent on the factor in the polynomial. Since the factor $$ (x+3) $$is raised to the power of 3, the zero at x = -3 has a multiplicity of 3.

Key Concepts

Polynomial Functions

In algebra, polynomial functions are expressions formed by the sum of powers of the variable, usually denoted as x, where the exponents are whole numbers, and the coefficients are real numbers. For example, \( y = ax^n + bx^{n-1} + \ldots + cx + d \) is a polynomial function of degree n, with a, b, c, and d being constants.

Polynomial functions represent a vast category of mathematical functions and can be relatively simple, like \( y = x^2 - 4 \), or more complex, like \( y = 2x^3 - 5x^2 + x - 1 \). Importantly, the degree of the polynomial tells us the maximum number of zeros (roots) a polynomial can have. It also provides insight into the function's graph, for instance, whether it is a straight line, a parabola, or has more turns and curves.

Multiplicity of Zeros

The multiplicity of a zero refers to how many times a particular zero occurs as a root of the polynomial. If a zero occurs n times, then it has a multiplicity of n. For example, for the polynomial function \( y = (x - 2)^3 \), the zero 2 has a multiplicity of 3.

Recognizing the multiplicity is key because it affects the graph of the polynomial function. A zero with even multiplicity means that the graph of the function touches the x-axis at that point and turns back, while a zero with odd multiplicity implies the graph will cross the x-axis. Knowing the multiplicity helps in understanding not only the roots but also the behavior of the polynomial's graph around those roots.

Solving Polynomial Equations

To resolve polynomial equations, like finding the zeros of \( y = (x+3)^3 \), you need to apply algebraic techniques to solve for x when y equals zero. These techniques vary depending on the complexity of the polynomial.

For simple polynomials, factoring might be sufficient. For quadratics, applying the quadratic formula helps. For higher degree polynomials, techniques like synthetic division, using the rational root theorem, or applying Descartes' Rule of Signs might be necessary. When polynomials are presented in factored form, as shown in our exercise, it becomes much easier to identify zeros by setting each factor equal to zero and solving for x, which represents where the graph of the polynomial will intersect the x-axis.