Problem 34 - College Algebra

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=(2 x+1)^{3}\left(9 x^{2}-6 x+1\right) $$

Step-by-Step Solution

Answer

Zeros: \(-\frac{1}{2}\) (multiplicity 3), \(\frac{1}{3}\) (multiplicity 2).

Step 1: Identify the Factors

The given function is \( f(x) = (2x+1)^3(9x^2 - 6x + 1) \). This is already factored into \((2x+1)^3\) and \(9x^2 - 6x + 1\). The zeros of the function will come from these factors.

Step 2: Solve \((2x+1)^3 = 0\)

Set the factor \((2x+1)^3\) equal to zero: 1. \((2x+1) = 0\) 2. Solve for \(x\): \(2x = -1\) \(x = -\frac{1}{2}\) The zero is \(x = -\frac{1}{2}\) with a multiplicity of 3.

Step 3: Solve \(9x^2 - 6x + 1 = 0\)

Use the quadratic formula to solve \(9x^2 - 6x + 1 = 0\). The formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 9\), \(b = -6\), and \(c = 1\). 1. Calculate \(b^2 - 4ac\): \((-6)^2 - 4(9)(1) = 36 - 36 = 0\) 2. Since the discriminant \(b^2 - 4ac = 0\), there is one real root (a double root): \(x = \frac{-(-6) \pm \sqrt{0}}{2 \times 9}\) \(x = \frac{6}{18}\) \(x = \frac{1}{3}\) The zero is \(x = \frac{1}{3}\) with a multiplicity of 2.

Step 4: Combine Results

The function \(f(x)\) has zeros at \(x = -\frac{1}{2}\) with multiplicity 3, and \(x = \frac{1}{3}\) with multiplicity 2.

Key Concepts

Multiplicity of Roots

The multiplicity of roots refers to the number of times a particular root is repeated in a polynomial equation. When you solve a polynomial, each solution or root can occur more than once, and each occurrence is counted individually.
If a root has a multiplicity greater than one, it means the polynomial touches or "bounces" off the x-axis at that point, rather than crossing it. This characteristic can significantly affect the graph's behavior at the root.

• A root with multiplicity 1 crosses the x-axis and changes sign.
• A root with an odd multiplicity greater than 1 touches the x-axis and turns back.
• A root with even multiplicity does not cross but bounces off the axis.

In our example, the zero at \( x = -\frac{1}{2} \) appears three times, indicating a multiplicity of 3. This means that on the graph of the function, it will touch the x-axis and change direction without crossing it.

Factored Form

Factoring a polynomial is a process of breaking it down into simpler, multiplied components known as factors. These factors can help us find the zeros or roots of the polynomial, which are values of \(x\) that make the entire polynomial equal to zero.
For our polynomial \( f(x) = (2x + 1)^3(9x^2 - 6x + 1) \), the expression is already presented in factored form. This makes finding its zeros easier, as it involves setting each distinct factor equal to zero.
The factored form allows us to directly identify the expressions we need to solve:

• \( (2x + 1)^3 \) contributes to one zero with a specific multiplicity.
• \( 9x^2 - 6x + 1 \) contributes to other zeros, possibly with different multiplicities.

This breakdown not only simplifies the solving process but also gives insights into the graph, where each factor's root represents points touching or cutting the x-axis, influenced by the roots' multiplicity.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the original exercise, the factor \( 9x^2 - 6x + 1 \) is a quadratic equation where \( a = 9 \), \( b = -6 \), and \( c = 1 \).
By applying the quadratic formula, we determined that the roots of this equation are real and equal, because the discriminant \( b^2 - 4ac \) is zero.
This results in a single solution, \( x = \frac{1}{3} \), that has a multiplicity of 2. The quadratic formula is particularly useful as it works for all cases, whether the roots are real or complex.

Discriminant Analysis

The discriminant is a component of the quadratic formula found under the square root and is given by \( b^2 - 4ac \). It plays a crucial role in determining the nature and number of roots a quadratic equation will have.
Here's how the discriminant can be interpreted:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, sometimes referred to as a double root, where the polynomial "touches" the x-axis.
• If \( b^2 - 4ac < 0 \), the equation has two complex roots, and the graph does not cross the x-axis.

In our exercise, the quadratic component \( 9x^2 - 6x + 1 \) has a discriminant of zero. This indicates the presence of one real root at \( x = \frac{1}{3} \) with a multiplicity of 2, as demonstrated by the double occurrence of this zero.