Problem 33 - College Algebra with Corequisite Support

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 (multiplicity 2) and -2 (multiplicity 2).

Step 1: Identify Factors of the Function

The given function is \( f(x) = x^2(x^2 + 4x + 4) \). We need to find zeros by setting the function equal to zero: \( x^2(x^2 + 4x + 4) = 0 \). This implies \( x^2 = 0 \) or \( x^2 + 4x + 4 = 0 \).

Step 2: Solve \( x^2 = 0 \)

Solve \( x^2 = 0 \), then take the square root of both sides: \( x = 0 \). The multiplicity of this zero is 2, since the factor \( x \) is squared, indicating it repeats twice.

Step 3: Factor \( x^2 + 4x + 4 \)

Factor the quadratic \( x^2 + 4x + 4 \) by finding two numbers that multiply to 4 and add to 4. The expression factors as \((x + 2)(x + 2) = (x+2)^2 \).

Step 4: Solve \( (x+2)^2 = 0 \)

Set \( (x + 2)^2 = 0 \) and solve for \( x \): \( x + 2 = 0 \) gives \( x = -2 \). The multiplicity of this zero is 2, since the factor \( (x+2) \) is squared.

Key Concepts

Zeros of a Polynomial

To find the zeros of a polynomial, start by setting the entire polynomial equal to zero. This process helps identify the values of the variable that make the equation true, effectively indicating where the graph of the polynomial touches or crosses the x-axis. For example, consider the polynomial function \( f(x) = x^2(x^2 + 4x + 4) \). Here, we set each factor equal to zero:

• \( x^2 = 0 \): Solving this gives \( x = 0 \).
• \( x^2 + 4x + 4 = 0 \): This can be factored and solved for \( x = -2 \).

Thus, the zeros of the polynomial \( f(x) = x^2(x^2 + 4x + 4) \) are \( x = 0 \) and \( x = -2 \). Each zero's location is crucial because it shows where the function changes sign.

Multiplicity of Roots

The multiplicity of a root in a polynomial tells us how many times a particular solution occurs. It corresponds to the power of the factor associated with that root. A higher multiplicity suggests the graph will touch, but not necessarily cross, the x-axis at that point. Let's look at these calculations for the polynomial \( f(x) = x^2(x^2 + 4x + 4) \):

• For \( x = 0 \), the factor is \( x^2 \). Here, the exponent 2 indicates the zero has a multiplicity of 2.
• For \( x = -2 \), the quadratic factors as \((x+2)^2\), giving this zero a multiplicity of 2 as well.

When the multiplicity is even, like 2, the graph touches the x-axis at the zero but does not cross it. If the multiplicity were odd, it would cross the axis. Therefore, \( f(x) \) touches the x-axis at both zeros but does not cross.

Factoring Quadratics

Factoring a quadratic is an essential skill to make solving polynomials easier, especially when searching for zeros. Factoring involves rewriting the quadratic in the form \((ax+b)(cx+d)\) or \((x+p)^2\) if it is a perfect square. Consider the quadratic part of the exercise, \(x^2 + 4x + 4\), which was factored as \((x+2)^2\):

• This form reveals if two numbers add up to the middle term (4) and multiply to the constant term (4).
• The terms \((x+2)(x+2)\) are equivalent to the square of a binomial, \((x+2)^2\).

Recognizing perfect squares speeds up the factorization process. Once in the factored form, it's simple to determine solutions by setting each binomial to zero. This method is efficient for solving equations and graphing polynomial roots.