Problem 260 - Precalculus

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=3 x^{4}+6 x^{3}+3 x^{2} $$

Step-by-Step Solution

Answer

Zeros: \( x = 0 \) (multiplicity 2), \( x = -1 \) (multiplicity 2).

Step 1: Factor out the Greatest Common Factor (GCF)

Look at the function \( f(x) = 3x^4 + 6x^3 + 3x^2 \). Each term has a common factor of \( 3x^2 \). Factor \( 3x^2 \) out of the polynomial: \[ f(x) = 3x^2(x^2 + 2x + 1) \]

Step 2: Factor the Quadratic Expression

Notice that the expression within the parentheses is a perfect square trinomial. That is, \( x^2 + 2x + 1 = (x + 1)^2 \). So, the function can be rewritten as: \[ f(x) = 3x^2(x + 1)^2 \]

Step 3: Find the Zeros of the Function

Set each factor in the equation \( 3x^2(x + 1)^2 = 0 \) equal to zero to find the zeros of the function. For \( 3x^2 = 0 \), solving gives \( x = 0 \). For \((x + 1)^2 = 0 \), solving gives \( x = -1 \). Thus, the zeros are at \( x = 0 \) and \( x = -1 \).

Step 4: Determine the Multiplicity of Each Zero

The multiplicity of a zero is the number of times that zero appears as a root of the polynomial equation. In the factored form \( 3x^2(x + 1)^2 \): - \( x = 0 \) has a multiplicity of 2 because of the \( 3x^2 \) factor. - \( x = -1 \) has a multiplicity of 2 because of the \( (x + 1)^2 \) factor.

Key Concepts

Greatest Common Factor

When working with polynomials, one of the first strategies to simplify an expression is to factor out the Greatest Common Factor (GCF). The GCF is the highest factor that divides all the terms in the polynomial without leaving any remainder.

For the polynomial \( f(x) = 3x^4 + 6x^3 + 3x^2 \) in the exercise, we need to examine each term.

• The terms are: \( 3x^4 \), \( 6x^3 \), and \( 3x^2 \).
• Each term includes the numerical factor 3.
• Additionally, \( x^2 \) is common to each term.

So, the GCF for this polynomial is \( 3x^2 \).

By factoring \( 3x^2 \) out, the expression simplifies to \( f(x) = 3x^2(x^2 + 2x + 1) \). This step reduces the complexity of the polynomial and prepares it for further factoring.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. Recognizing perfect squares can help simplify polynomials effectively.

The expression we obtained after factoring out the GCF is \( x^2 + 2x + 1 \). This is a perfect square trinomial.

• It can be rewritten as \((x + 1)^2 \).
• The general form of a perfect square trinomial is \( (a + b)^2 = a^2 + 2ab + b^2 \).
• Here, \( a = x \) and \( b = 1 \), making \( a^2 = x^2 \), \( 2ab = 2 \times x \times 1 = 2x \), and \( b^2 = 1 \).

Understanding how to recognize and factor perfect square trinomials is a powerful tool for simplifying polynomial expressions.

Multiplicity of Zeros

Multiplicity refers to the number of times a specific zero appears as a root of a polynomial equation. Recognizing this is essential when solving polynomial equations, as it affects the curve's intersections on the graph.

From the exercise, the factored form of the polynomial is \( 3x^2(x + 1)^2 \). This helps us easily identify the zeros and their multiplicities:

• For \( 3x^2 = 0 \), the zero is \( x = 0 \) and it has a multiplicity of 2. This result is due to the factor \( x^2 \).
• For \((x + 1)^2 = 0 \), the zero is \( x = -1 \) and it also has a multiplicity of 2, attributed to the factor \((x + 1)^2 \).

A zero of higher multiplicity tends to "touch" the x-axis at its corresponding value but does not cross it, indicating a more complex interaction at these points.