Problem 261 - Precalculus

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=4 x^{5}-12 x^{4}+9 x^{3} $$

Step-by-Step Solution

Answer

Zeros are 0 (multiplicity 3) and \(\frac{3}{2}\) (multiplicity 2).

Step 1: Factor Out the Greatest Common Factor (GCF)

First, identify the greatest common factor of all terms in the polynomial. The GCF of \(4x^5\), \(-12x^4\), and \(9x^3\) is \(x^3\). Factor it out: \[ f(x) = x^3 (4x^2 - 12x + 9) \].

Step 2: Factor the Quadratic Expression

Now focus on factoring the quadratic \(4x^2 - 12x + 9\). Use the factoring technique for quadratic equations. Begin by checking if it can be factored into two binomials: \[ 4x^2 - 12x + 9 = (2x - 3)(2x - 3) = (2x - 3)^2 \]. This shows that \(4x^2 - 12x + 9\) is a perfect square trinomial.

Step 3: Find the Zeros of the Polynomial

Set the factored form of the polynomial equal to zero to find the zeros: \[ x^3 (2x-3)^2 = 0 \]. This equation gives us two distinct scenarios. First, \(x^3 = 0\) which leads to \(x = 0\). Second, \((2x-3)^2 = 0\) which can be further solved as \(2x - 3 = 0\), resulting in \(x = \frac{3}{2}\).

Step 4: Determine the Multiplicity of Each Zero

The zero \(x = 0\) arises from \(x^3\) and has a multiplicity of 3, because \(x^3 = (x)(x)(x)\). The zero \(x = \frac{3}{2}\) arises from \((2x-3)^2\) and has a multiplicity of 2, as seen from the squared factor.

Key Concepts

Factoring Polynomials

Factoring polynomials is a method used to simplify expressions and solve polynomial equations efficiently. Polynomials can often be written as a product of simpler polynomials or other expressions. When factoring polynomials, you break down a given polynomial into its simplest components, known as factors. This process is essential, especially when solving equations to find zeros or roots.

• Start by identifying if there is a common factor that you can factor out from all terms.
• Check for recognizable patterns or expressions, like perfect square trinomials or difference of squares.
• Use various methods such as grouping, trinomial factoring, and special formulas to factor completely.

In the exercise example, the polynomial is initially factored by looking for a Greatest Common Factor (GCF), and then factoring the resulting quadratic expression.

Zero Multiplicity

A zero of a polynomial function is a value of the variable that makes the function equal to zero. Zero multiplicity refers to the number of times a particular zero occurs. Essentially, it indicates how many times a factor is repeated in the factored form of the polynomial.

• A zero with a multiplicity of 1 is called a simple or single root.
• A zero with a multiplicity greater than 1 means the corresponding factor is raised to an additional power.

In the given solution, the polynomial has two zeros:

• Zero at \(x = 0\) with a multiplicity of 3 because it arises from \(x^3\).
• Zero at \(x = \frac{3}{2}\) with a multiplicity of 2 due to the squared factor \((2x-3)^2\).

These multiplicities provide insight into the behavior of the graph of the polynomial near these zeros.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that can divide each term in the polynomial without a remainder. Identifying the GCF is often the first step in the factoring process, making it easier to manage and simplify expressions.To find the GCF:

• Identify the greatest coefficient common to each term.
• Determine the smallest power of each variable that appears in each term.
• Multiply these together to determine the GCF.

In our example, the GCF of the terms \(4x^5\), \(-12x^4\), and \(9x^3\) was \(x^3\). Factoring out the GCF simplifies the polynomial into \(x^3(4x^2 - 12x + 9)\), allowing for further factorization and solution of the polynomial equation.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, generally having the form \(ax^2 + bx + c\). These expressions can often be factored into binomials, especially when they fit special patterns.

• If a quadratic can be expressed as \((px + q)^2\), it is a perfect square trinomial.
• For other quadratics, methods such as the quadratic formula, completing the square, or factoring can be employed.

In the resolved exercise, the quadratic expression \(4x^2 - 12x + 9\) was shown to be a perfect square trinomial. It factored to \((2x-3)^2\), indicating that this quadratic expression has a repeated root.