Problem 41 - College Algebra with Corequisite Support

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

The zeros are at \(x = 0\) with multiplicity 6, and \(x = \frac{2}{3}\) with multiplicity 2.

Step 1: Factor the Common Terms

Begin by factoring the expression inside the polynomial. Notice that each term in the polynomial inside the brackets has a common factor of \(x^2\). So, factor \(x^2\) out from \(9x^4 - 12x^3 + 4x^2\) to get \(x^2(9x^2 - 12x + 4)\).

Step 2: Rewrite the Function with Factoring

The expression for \(f(x)\) now becomes: \[ f(x) = 4x^4 \times x^2 \times (9x^2 - 12x + 4) = 4x^6(9x^2 - 12x + 4) \] This reflects the entire polynomial function in factored form.

Step 3: Solve for Zeros of the Factored Terms

To find the zeros of \(f(x)\), solve each part of the factorized expression. First, consider \(4x^6 = 0\), which gives \(x = 0\).

Step 4: Determine Multiplicity of Zero at x = 0

The zero \(x = 0\) comes from the term \(4x^6\), so it has a multiplicity of 6. This means that the root \(x = 0\) occurs 6 times.

Step 5: Solve the Quadratic for Additional Zeros

Now, focus on the remaining quadratic factor \(9x^2 - 12x + 4 = 0\). This can be solved using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 9\), \(b = -12\), and \(c = 4\).

Step 6: Calculate Discriminant and Roots

Calculate the discriminant: \[ b^2 - 4ac = (-12)^2 - 4 \times 9 \times 4 = 144 - 144 = 0 \]Since the discriminant is 0, there is one real double root. Now, calculate:\[ x = \frac{-(-12) \pm 0}{2 \times 9} = \frac{12}{18} = \frac{2}{3} \]

Step 7: Determine Multiplicity of Root at x = 2/3

The root \(x = \frac{2}{3}\) has a multiplicity of 2, owing to the double root indicated by the zero discriminant.

Key Concepts

Multiplicity of Roots

Understanding the multiplicity of roots is crucial in determining how many times a particular root appears in a polynomial. If a root appears more than once, it is said to have a multiplicity greater than one.
A simple example is when a factor like \( (x - a)^n \) occurs in the factorization of a polynomial. In this case, the root \( x = a \) has a multiplicity of \( n \).
In our original exercise, the root \( x = 0 \) has a multiplicity of 6, coming from the term \( 4x^6 \). This means \( x = 0 \) is a root repeated 6 times. In practical terms, this impacts the shape of the graph near \( x = 0 \), where it will "kiss" the x-axis instead of crossing it. Another root \( x = \frac{2}{3} \) has a multiplicity of 2, known as a double root, which also means the graph "touches" the axis at this point instead of crossing.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials which, when multiplied together, give the original polynomial.
This is an essential skill because it simplifies the process of finding roots of the polynomial. For the given function \( f(x) = 4x^4(9x^4 - 12x^3 + 4x^2) \)
, we began by identifying common factors. We factored out \( x^2 \) from \( 9x^4 - 12x^3 + 4x^2 \), resulting in this: \( f(x) = 4x^6(9x^2 - 12x + 4) \).

• This multiple-step factoring allows us to easily identify zeros and their multiplicities.
• Each factor corresponds to a potential zero of the polynomial.

Factoring can sometimes be complex, but it simplifies solving, allowing us to systematically address each polynomial factor.

Quadratic Formula

The quadratic formula provides a method for finding the roots of quadratic equations. It's a formula that applies when you have a quadratic equation of the form \( ax^2 + bx + c = 0 \).
In the formula, given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), the letters \( a, b, \text{ and } c \) represent the coefficients of the terms in the quadratic equation.
Using our exercise:

• \( a = 9 \)
• \( b = -12 \)
• \( c = 4 \)

Plugging these values into the quadratic formula helps us find the roots of \( 9x^2 - 12x + 4 = 0 \).
The formula gives us roots directly, and if calculated properly, it is foolproof for any quadratic!

Discriminant Calculation

The discriminant is an important part of the quadratic formula, denoted as \( b^2 - 4ac \).
It helps determine the nature of the roots without actually solving the quadratic equation.
Depending on the value:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root with multiplicity two, known as a double root.
• If \( b^2 - 4ac < 0 \), there are no real roots.

In the given exercise, the discriminant is \( 144 - 144 = 0 \), leading to one real double root \( x = \frac{2}{3} \).
This tool helps quickly ascertain the root structure of quadratics!