Problem 30 - 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. \(f(x)=x^{3}+4 x^{2}+4 x\)

Step-by-Step Solution

Answer

The zeros of the function are \(x=0\) with multiplicity 1 and \(x=-2\) with multiplicity 2. The graph crosses the x-axis at \(x=0\) and touches the x-axis and turns around at \(x=-2\).

Step 1: Find the Zeros

To find the zeros of the function \(f(x)=x^{3}+4 x^{2}+4 x\), set the equation equal to 0, i.e, solve \(x^{3}+4 x^{2}+4 x=0\). This equation can be factored as \(x*(x+2)^2=0\). Setting each factor equal to zero gives the solutions \(x=0, x=-2, x=-2\).

Step 2: Determine the Multiplicity

The zero \(x=0\) occurs once, so it has multiplicity 1. The zero \(x=-2\) occurs twice, so it has multiplicity 2.

Step 3: Graph Behavior at Zeros

Given that the zero \(x=0\) has an odd multiplicity, the graph crosses the x-axis at \(x=0\). The zero \(x=-2\) has an even multiplicity. Hence, the graph touches the x-axis and turns around at \(x=-2\).

Key Concepts

Multiplicity of Zeros

When we talk about zeros of a polynomial, we refer to the values of \(x\) for which the polynomial equals zero. Each zero can have a multiplicity, which can be thought of as the number of times a particular zero is repeated as a factor in the polynomial. In simple terms, multiplicity tells us how many times a root is repeated. For example, in the equation \((x+2)^2\), the zero \(x = -2\) has a multiplicity of 2, indicating that this zero is repeated twice in the factorization.Understanding the multiplicity of zeros is crucial because it helps us predict the behavior of the graph at each zero. If the zero has odd multiplicity, the graph will cross the x-axis at this point. If the multiplicity is even, the graph will touch the x-axis and turn around without crossing it.

• Zero with odd multiplicity: graph crosses the x-axis.
• Zero with even multiplicity: graph touches and turns around at the x-axis.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler terms, known as "factors," which multiplied together give back the original polynomial. This is analogous to breaking down a number into its prime factors.For the polynomial \(f(x)=x^{3}+4x^{2}+4x\), the first step is to factor out the greatest common factor, \(x\). This yields:\[f(x) = x(x^2 + 4x + 4)\]Next, factor the quadratic \((x^2 + 4x + 4)\) further:\[x^2 + 4x + 4 = (x + 2)^2\]Combining these factors, we have:\[f(x) = x(x + 2)^2\]Factoring not only helps in finding the zeros but also makes it easier to determine the multiplicity of each zero. In this case, \(x = 0\) is a factor with multiplicity 1, and \(x = -2\) is a factor with multiplicity 2.

Graph Behavior at Zeros

Understanding how a graph behaves at its zeros is essential for sketching the polynomial’s graph accurately. The multiplicity of each zero significantly affects this behavior.If a zero has an odd multiplicity, the graph of the polynomial will cross the x-axis at that zero. Conversely, if the zero has an even multiplicity, the graph will touch the x-axis at that zero but will "turn around" instead of crossing it.In our example of \(f(x) = x(x + 2)^2\):

• At \(x = 0\), with multiplicity 1 (odd), the graph crosses the x-axis.
• At \(x = -2\), with multiplicity 2 (even), the graph touches the x-axis and turns around.

Knowing this behavior helps when sketching graphs and interpreting the behavior of polynomial functions in practical situations.