Problem 33 - College Algebra

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x^{2}\left(x^{2}+4 x+4\right) $$

Step-by-Step Solution

Answer

The zeros are 0 with multiplicity 2 and -2 with multiplicity 2.

Step 1: Factor the Expression

The function given is \[ f(x) = x^2(x^2 + 4x + 4) \]Firstly, we notice that the inner quadratic expression inside the parentheses can be factored further. We need to factor \(x^2 + 4x + 4\). This expression can be rewritten as \[ (x+2)(x+2) \] or \( (x+2)^2 \). So the complete factorization of \(f(x)\) would be:\[ f(x) = x^2(x+2)^2 \].

Step 2: Identify the Zeros

To find the zeros of \(f(x)\), we set the function equal to zero:\[ x^2(x+2)^2 = 0 \]The equation is zero when either \(x^2=0\) or \((x+2)^2=0\). Solving these gives us zeros:1. \(x^2=0\) leads to \(x=0\).2. \((x+2)^2=0\) leads to \(x=-2\).

Step 3: Determine the Multiplicity

Multiplicity refers to how many times a particular zero appears in the factorization of the function:1. The zero \(x=0\) comes from the factor \(x^2\), which means the multiplicity is 2.2. The zero \(x=-2\) comes from the factor \((x+2)^2\), which also has a multiplicity of 2.

Key Concepts

Finding Zeros

Finding the zeros of a polynomial function involves solving the equation where the polynomial is set to zero. In essence, you are looking for the values of the variable that make the whole function equal to zero. For the function \[ f(x) = x^2(x^2 + 4x + 4), \]our task is simple once it is factored.

• The expression can be broken down into two parts: \(x^2\) and \( (x+2)^2 \).
• Each part gives us different zeros.

For \(x^2 = 0\), solving for \(x\) gives \(x = 0\). Similarly, for \((x+2)^2 = 0\), solving for \(x\) gives \(x = -2\). These values—0 and -2—are the zeros of the polynomial. Finding the zeros is crucial as it tells us where the function crosses or touches the x-axis on a graph. This step is an essential part of analyzing the behavior of polynomial functions.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler terms, or factors, that when multiplied together give you the original polynomial. This process simplifies the problem of finding zeros.

• With the given polynomial \( f(x) = x^2(x^2 + 4x + 4) \), our aim was to factor the expression fully.
• The quadratic \(x^2 + 4x + 4\) factored to \((x+2)^2\), which can be verified through expanding \((x+2)(x+2)\) back to the original quadratic expression.

The complete factorization is: \[ f(x) = x^2(x+2)^2 \] This form allows us to see clearly what our zeros are and their multiplicities. By factoring, we simplify the process of finding zeros and gain insight into the function's graph. Factoring makes complex problems easier to handle and is an important technique in algebra.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. In simpler terms, it's how often each zero is repeated as a factor of the polynomial. Understanding multiplicity helps us predict the behavior of the graph of the polynomial around its zeros.

• In our example, the zero \(x=0\) comes from the factor \(x^2\).
• This means it appears twice, thus it has a multiplicity of 2.
• Similarly, the zero \(x=-2\) comes from the factor \((x+2)^2\), also appearing twice and having a multiplicity of 2.

Zeros with an odd multiplicity will cause the graph to cross the x-axis at the zero, while zeros with an even multiplicity will "bounce" off the axis without crossing it. This knowledge is particularly useful in sketching graphs and understanding polynomial behavior.