Problem 26 - Algebra and Trigonometry

Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x@axis, or touches the x@axis and turns around, at each zero. \(f(x)=3(x+5)(x+2)^{2}\)

Step-by-Step Solution

Answer

The zeros of the function are -5 with multiplicity 1 and -2 with multiplicity 2. The graph cross the axis at x=-5, and touches the axis and turns around at x=-2.

Step 1: Identify and List All Factors

The provided polynomial function is \(f(x)=3(x+5)(x+2)^2\) . It has two individual factors, which are \(x+5\) and \(x+2\)

Step 2: Find the Zeros

To find zeros, we'll set each factor equal to 0 and solve for x. So, for the first factor \(x+5=0\), we find that \(x=-5\). For the second factor \(x+2=0\), we get \(x=-2\)

Step 3: Find the Multiplicity

The multiplicity of a root is determined by the exponent of the factor in the factorized function. For the root -5, the corresponding factor (x+5) has no visible exponent, which means it's to the power of 1. So, multiplicity of -5 is 1. For the root -2, the exponent of factor (x+2) is 2, hence, multiplicity of -2 is 2

Step 4: Determine the Behavior of the Graph at Each Zero

The behaviour of the graph at each zero is determined by their multiplicities. If the multiplicity is odd, the graph will cross the x-axis at the zero, and if it's even, the graph just touches the x-axis and turns around at the zero. The root -5 has an odd multiplicity (1), so the graph crosses axis at x=-5. The root -2 has an even multiplicity (2), so the graph touches at x=-2 and turns around

Key Concepts

Multiplicity of Zeros

When discussing the zeros of a polynomial function, one essential aspect is understanding the concept of the multiplicity of each zero. The multiplicity of a zero refers to the number of times a particular zero appears in the factorization of the polynomial.
For example, consider the function \( f(x) = 3(x+5)(x+2)^2 \). We start by looking at the factors: \( x+5 \) and \( (x+2)^2 \). Each factor's exponent indicates the multiplicity of its corresponding zero.

• If a factor is written as \((x-a)^n\), the zero \(x=a\) appears \(n\) times, giving it a multiplicity of \(n\).
• In our polynomial, the factor \((x+5)\) is to the power of 1, indicating a multiplicity of 1. So, \(x=-5\) is a zero with multiplicity 1.
• The factor \((x+2)^2\) shows up with an exponent of 2, thus \(x=-2\) is a zero with multiplicity 2.

Understanding multiplicity helps us not only with finding zeros but also in exploring how a graph behaves at these zeros.

Graph Behavior at Intercepts

The behavior of a polynomial graph at its intercepts provides significant insight into how the graph looks near the x-axis. The key factor influencing this behavior is the multiplicity of each zero.
For zeros with:

• **Odd Multiplicity**: The graph will cross the x-axis at these points.
• **Even Multiplicity**: The graph only touches the x-axis and then turns around, creating a kind of "bounce".

Let's look at our function \( f(x) = 3(x+5)(x+2)^2 \). For \( x = -5 \), we have a zero with odd multiplicity (1), meaning the graph crosses the x-axis at \(x = -5\).
For \( x = -2 \), a zero with even multiplicity (2), the graph touches the x-axis at \(x = -2\) and then turns back, resembling a gentle touch and retreat.
This crossing or bouncing helps in sketching a rough idea of the curve of a polynomial without needing detailed computation.

Polynomial Factorization

Factorization is a crucial step in solving polynomial equations. It involves breaking down a polynomial into simpler "factors," making it easier to find zeros and understand the polynomial's properties.
For the function \( f(x) = 3(x+5)(x+2)^2 \), factorization is already given:

• The constant term "3" is a common factor but doesn’t impact the zeros.
• Each term \( (x+5) \) and \( (x+2)^2 \) represents the polynomial’s factors. These factors help deduce the zeros: \(x = -5\) and \(x = -2\) respectively.

The simplicity of factorization makes it a preferred method compared to other means of finding zeros like graphing, especially for higher-degree polynomials.
Once a polynomial is factorized, determining its zeros and analyzing graph behavior becomes straightforward. Always aim to express polynomials in their simplest factorized form for easy analysis and problem-solving.