Problem 41 - Precalculus with Calculus Previews

Question

Find the zeros of the given polynomial function \(f .\) State the multiplicity of each zero. $$ f(x)=\left(9 x^{2}-4\right)^{2} $$

Step-by-Step Solution

Answer

The zeros are \( x = \frac{2}{3} \) and \( x = -\frac{2}{3} \), each with multiplicity 2.

Step 1: Set the Function Equal to Zero

To find the zeros of the polynomial function, we need to set the function equal to zero: \( f(x) = (9x^2 - 4)^2 = 0 \). This means the expression inside the parentheses must also be zero for \( f(x) \) to be zero.

Step 2: Solve the Inner Expression for Zero

Solve \( 9x^2 - 4 = 0 \). Add 4 to both sides to isolate the quadratic term: \( 9x^2 = 4 \).

Step 3: Solve for x

Divide both sides by 9 to solve for \( x^2 \): \( x^2 = \frac{4}{9} \). Take the square root of both sides: \( x = \pm \frac{2}{3} \).

Step 4: State the Multiplicity

The original function \( f(x) = (9x^2 - 4)^2 \) indicates that each solution \( x = \frac{2}{3} \) and \( x = -\frac{2}{3} \) has a multiplicity of 2 since the inner polynomial \( 9x^2 - 4 \) is squared.

Key Concepts

Quadratic Equation

Quadratic equations are a type of polynomial equation specifically where the highest power of the variable, typically x, is squared. They generally come in the form of \( ax^2 + bx + c = 0 \). Solving a quadratic equation means finding the value(s) for x that make the equation true. These values are often referred to as the "solutions" or "roots" of the equation.

In our exercise, we specifically look at an expression \( 9x^2 - 4 \), which resembles the standard quadratic form. Solving such equations often involves using straightforward methods such as:

• Factoring: If the quadratic expression can be factored to a form like \((d)(e) = 0\), we find the roots by setting each factor equal to zero.
• Using the quadratic formula: Applied when factoring isn't feasible, this formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides a direct route to the solutions.
• Completing the square: Another approach is rearranging the equation into a perfect square form and then solving.

The equation \( 9x^2 = 4 \) results from setting our quadratic equal to zero. By manipulating it algebraically, such as dividing by the coefficient of \( x^2 \), we can eventually solve for x.

Multiplicity of Zeros

The concept of multiplicity in polynomials refers to how many times a particular root, or zero, appears in the solution. In other words, if a root occurs "twice", it has a multiplicity of 2, and if "thrice", then 3, and so on.

In the original exercise, the entire expression \((9x^2 - 4)^2 = 0\) makes it evident that each of its solved roots, \( x = \frac{2}{3} \) and \( x = -\frac{2}{3} \), has a multiplicity of 2. This is because the squared notation indicates that the sequence will repeat itself twice whenever the factor \( 9x^2 - 4 \) results in zero.

Understanding multiplicity is helpful because:

• It tells you how the polynomial's graph behaves at each of the roots.
• A root with an even multiplicity indicates that the graph will touch the x-axis at this zero but not cross it.
• A root with an odd multiplicity means the graph crosses the x-axis at the zero.

Recognizing multiplicity provides insight into the solution's structure and the polynomial's behavior over the graph.

Solving Equations

Solving equations involves finding the values that make the equation valid, meaning everything balances correctly. For polynomial functions, this typically means solving for the variable, often noted as "x", where the polynomial equals zero.

Let's look at a condensed step-by-step process relevant to the exercise:

• **Set the Equation to Zero**: Determine when the polynomial function equals zero, making it necessary to manipulate it into a zero-at-right format, like \( f(x) = 0 \).
• **Isolate the Variable**: If the polynomial is quadratic or otherwise factored form, separate its primary components in order to simplify and identify the dependent variable's values.
• **Solve Algebraically**: Apply algebraic methods like factoring, using the quadratic formula, or even completing the square to solve for x, often leading to several steps, which may include adding, subtracting, dividing, and even squaring both sides or square-rooting.
• **Check for Multiplicity**: Analyze the function's original algebraic form for expressions using powers, like squared factors. This helps in determining each zero with its associated multiplicity.

By carefully progressing through these steps, the solutions flow logically, making complex equations more approachable and less daunting for students.