Problem 41 - College Algebra

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=4 x^{4}\left(9 x^{4}-12 x^{3}+4 x^{2}\right) $$

Step-by-Step Solution

Answer

Zeros are \( x = 0 \) (mult. 6) and \( x = \frac{2}{3} \) (mult. 2).

Step 1: Factor the Function

First, we factor the function completely to analyze the roots. The given function is as follows: \[ f(x) = 4x^4(9x^4 - 12x^3 + 4x^2). \] Factor out common terms within the polynomial. Notice that the polynomial inside can be further factored. Start by factoring out \( x^2 \): \[ 9x^4 - 12x^3 + 4x^2 = x^2(9x^2 - 12x + 4). \] Thus, we have: \[ f(x) = 4x^4x^2(9x^2 - 12x + 4). \] This simplification gives us: \[ f(x) = 4x^6(9x^2 - 12x + 4). \] Now, attempt further factorization of \( 9x^2 - 12x + 4 \) to find more roots.

Step 2: Solve the Quadratic Equation

To factor further or find zeros, let's solve the quadratic equation \[ 9x^2 - 12x + 4 = 0. \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 9 \), \( b = -12 \), and \( c = 4 \).Calculate the discriminant: \[ b^2 - 4ac = (-12)^2 - 4 \cdot 9 \cdot 4 = 144 - 144 = 0. \]Since the discriminant is 0, the quadratic has exactly one real double root. Calculate:\[ x = \frac{-(-12) \pm \sqrt{0}}{2 \cdot 9} = \frac{12}{18} = \frac{2}{3}. \]

Step 3: Identify Zeros and Their Multiplicity

Combine all the information from the factored form of the function. From Step 1, we have:- A zero at \( x = 0 \) with multiplicity 6 due to \( 4x^6 \).From Step 2, we found:- A zero at \( x = \frac{2}{3} \) with multiplicity 2.Thus, the zeros are \( x = 0 \) with multiplicity 6 and \( x = \frac{2}{3} \) with multiplicity 2.

Key Concepts

Multiplicity of Roots

When we talk about the roots of a polynomial, we often need to consider not only their values but also their multiplicity. The **multiplicity of a root** tells us how many times a particular root appears in the polynomial. For instance, if a root appears twice, it is known as a double root, meaning its multiplicity is 2. The concept of multiplicity is important because it affects the shape of the graph at those points. A root with a higher multiplicity will typically cause the graph to be flatter at that point.

In the provided solution, we identified zeros and their multiplicity:

• For the expression \(4x^6\), we determined a root at \(x = 0\) with a multiplicity of 6. This means that the root \(x = 0\) appears 6 times.
• The quadratic factor \(9x^2 - 12x + 4\) results in a root at \(x = \frac{2}{3}\) with a multiplicity of 2, signifying the root appears twice.

Understanding multiplicity helps in sketching the graph and knowing the behavior of the polynomial around these roots.

Factoring Polynomials

Factoring is an essential step in finding the zeros of a polynomial. **Factoring polynomials** involves breaking down a polynomial into simpler "factors" or components, which multiply together to give the original polynomial. It simplifies the equation and makes finding roots much easier. There are several methods to factor polynomials, including:

Solving Quadratic Equations

Solving quadratic equations is a cornerstone in algebra. Quadratic equations take the form \(ax^2 + bx + c = 0\). To solve them, you can use several techniques such as factoring, completing the square, or employing the quadratic formula. In this exercise, we used the quadratic formula to find the roots of the quadratic part of the polynomial.

The quadratic formula is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula uses specific coefficients \(a, b,\) and \(c\) from the quadratic equation. By substituting these values, we can find the roots.

In this specific problem, using the quadratic formula with \(a = 9\), \(b = -12\), and \(c = 4\) yielded one repeated root at \(x = \frac{2}{3}\), due to the discriminant being zero.

Discriminant of Quadratic Equation

The **discriminant** of a quadratic equation provides valuable insight into the nature of the roots. The discriminant is denoted by \(b^2 - 4ac\) and is derived from the quadratic formula. It helps us determine:

• If there are two distinct real roots (when discriminant > 0).
• If there is exactly one real root (when discriminant = 0), known as a double root.
• If there are no real roots and the roots are complex (when discriminant < 0).

In our exercise, the discriminant of the quadratic \(9x^2 - 12x + 4\) was calculated as zero,indicating a double root at \(x = \frac{2}{3}\).

Understanding the discriminant allows us not only to find the number of roots but also to anticipate the nature of those roots, which is essential for sketching the graph or solving related problems.