Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x\left(4 x^{2}-12 x+9\right)\left(x^{2}+8 x+16\right) $$

Step-by-Step Solution

Answer

Zeros: 0 (1), \(\frac{3}{2}\) (2), -4 (2)

Step 1: Identify Roots of Linear Factors

The function is given as a product of three factors: \(f(x) = x(4x^2 - 12x + 9)(x^2 + 8x + 16)\). First, consider the linear factor \(x\). The root for \(x\) is at \(x = 0\) with a multiplicity of 1, because the factor appears once in the expression.

Step 2: Solve the Quadratic Equation 1

Next, solve the quadratic equation \(4x^2 - 12x + 9 = 0\). This can be factored into \((2x - 3)^2 = 0\) by recognizing it as a perfect square. Solving \(2x - 3 = 0\), we find the root \(x = \frac{3}{2}\) with multiplicity 2 (because the factor \((2x-3)\) appears squared).

Step 3: Solve the Quadratic Equation 2

Now solve the quadratic equation \(x^2 + 8x + 16 = 0\). This can be factored as \((x + 4)^2 = 0\) by recognizing it as a perfect square. Solving \(x + 4 = 0\), we find the root \(x = -4\) with multiplicity 2 (because the factor \((x + 4)\) appears squared).

Step 4: Summarize the Zeros and Their Multiplicities

The function \(f(x)\) has three distinct zero values. The roots are \(x = 0\) with multiplicity 1, \(x = \frac{3}{2}\) with multiplicity 2, and \(x = -4\) with multiplicity 2. Each zero is derived from the respective factor of the polynomial.

Key Concepts

Multiplicity of Roots

When finding the zeros of a polynomial function, it is important to consider the multiplicity of each root. The term "multiplicity" refers to the number of times a particular root occurs in a polynomial equation. For instance, a quadratic factor that is repeated twice in a polynomial shows a root with a multiplicity of 2.

Multiplicity is important because it affects the graph's behavior at the root. If the root's multiplicity is even, the graph touches the x-axis but doesn't cross it. If it's odd, the graph will cross the x-axis at that point.

• Single occurrence: The graph crosses the x-axis at the root.
• Double occurrence: The graph touches the x-axis and bounces back.
• Triple occurrence: The graph crosses and flattens at the x-axis.

Understanding multiplicity aids in sketching polynomial graphs and solving various algebraic problems.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which can be expressed in the standard form \(ax^2 + bx + c = 0\). Solving quadratic equations is crucial for finding the roots of polynomial functions, particularly when examining factors.

There are several methods to solve quadratic equations:

• Factoring
• Completing the square
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Quadratics often result in two solutions because they describe the behaviour of a parabola.
For example, the quadratic \(4x^2 - 12x + 9 = 0\), solves into \((2x-3)^2 = 0\), offering a single solution \(x = \frac{3}{2}\) but with a multiplicity of 2.

Factoring Polynomials

Factoring is a method used to reduce a polynomial expression into simpler components, which can make solving for zeros much easier. Essentially, factoring rewrites a polynomial as a product of its factors.

When factoring polynomials, it's often useful to look for recognizable patterns, such as the difference of squares or perfect square trinomials.

• Look for common factors in all terms, if they exist.
• Identify and factor perfect squares: e.g., \(a^2 + 2ab + b^2 = (a + b)^2\).
• Use grouping techniques for higher degree polynomials.

For example, the polynomial \(f(x) = x(4x^2 - 12x + 9)(x^2 + 8x + 16)\) is already partially factored. Recognizing the inner quadratics as perfect squares simplifies finding the zeros and makes it easier to determine the multiplicity of each root.