Problem 32 - College Algebra

Question

In Exercises \(27-34,\) 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 \(f(x)=x^{3}+4 x^{2}+4x\) 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: Factor the polynomial

The given function is \(f(x)=x^{3}+4 x^{2}+4x\). By factoring out the common term \(x\), the equation becomes \( f(x)= x*(x^{2}+4x+4)\). This equation can be easily factored to \(f(x)=x*(x+2)^{2}\).

Step 2: Find the zeros of the function

The zeros of the function are the values of \(x\) that make \(f(x)=0\). Therefore, solving the equation \(x*(x+2)^{2}=0\) gives the roots \(x=0\) and \(x=-2\). However, \(x=-2\) has a multiplicity of 2 because it is a solution to the equation two times.

Step 3: Determine the behavior of the function at the zeros

For \(x=0\), the function just touches the x-axis and then turns around. This is because \(x=0\) has a multiplicity of 1, which is odd, indicating that the graph crosses the x-axis at \(x=0\). For \(x=-2\), the function touches the x-axis and then turns around. The reason for this is that \(x=-2\) has a multiplicity of 2, which is even, indicating that the graph touches the x-axis and turns around at \(x=-2\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Each term in a polynomial has the form of a constant multiplied by the variable raised to a non-negative integer power. For instance, in the given problem, the function
\( f(x) = x^3 + 4x^2 + 4x \) is a polynomial function of degree 3, as the highest power of \( x \) is 3.

The degree of the polynomial dictates the possible number of zeros and extremas it may have. In simple terms, a polynomial of degree \( n \) can have at most \( n \) real zeros, which are the points where the polynomial intersects with the \( x \)-axis when graphed. These zeros are crucial for understanding the behavior of the function, such as where it increases or decreases, and are foundational in graphing the overall shape of the polynomial function.

Factoring Polynomials

Factoring polynomials is a key skill in algebra that involves breaking down the polynomial into simpler terms that, when multiplied together, give you the original polynomial. This process is akin to finding what numbers multiply together to give another number. For instance, if you have the polynomial \( x^2 + 5x + 6 \), it can be factored into \( (x + 2)(x + 3) \).

In the exercise, the polynomial \( f(x) = x^3 + 4x^2 + 4x \) is factored by first taking out the common factor of \( x \), resulting in \( x(x^2 + 4x + 4) \). Further, since \( x^2 + 4x + 4 \) is a perfect square, it can be factored again into \( (x + 2)^2 \). Thus, the fully factored form of the polynomial is \( x(x + 2)^2 \). Factoring helps to find the zeros of a polynomial function, which are the solutions to the equation obtained by setting the polynomial equal to zero.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular solution appears in a factored polynomial. If a factor is repeated, each replication is counted in the multiplicity. Simply put, it indicates how many times the function 'touches' or crosses the \( x \)-axis at that zero position when graphed.

Zeros with an odd multiplicity will result in the graph crossing the \( x \)-axis, signaling a change in direction from positive to negative or vice versa. In contrast, zeros with an even multiplicity mean the graph merely 'kisses' or touches the \( x \)-axis and then turns back, without crossing it.

In our example, the zero \( x = -2 \) has a multiplicity of 2 since the factor \( (x + 2) \) is squared. This implies that at \( x = -2 \), the graph of the polynomial touches the \( x \)-axis and turns around. Meanwhile, \( x = 0 \) has a multiplicity of 1, which indicates the polynomial's graph will cross the \( x \)-axis at this point. Understanding multiplicity helps with sketching accurate graphs of polynomial functions and is essential in various calculus applications.