Problem 39 - College Algebra with Corequisite Support

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: \(x = 0\) (multiplicity 3), \(x = \frac{3}{2}\) (multiplicity 2).

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

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

Step 2: Factor the Quadratic Expression

Now, focus on factoring the quadratic expression \(4x^2 - 12x + 9\). This can be factored by looking for two numbers that multiply to \(4 \times 9 = 36\) and add to \(-12\). The numbers \(-6\) and \(-6\) work: \[4x^2 - 12x + 9 = (2x - 3)(2x - 3)\]Rewrite the function with the factors: \[f(x) = x^3 (2x - 3)^2\]

Step 3: Identify the Zeros

The factored expression \(x^3(2x-3)^2\) suggests zeros at \(x=0\) and \(x=\frac{3}{2}\). These are found by setting each factor equal to zero: 1. \(x^3 = 0\) gives \(x = 0\).2. \((2x - 3)^2 = 0\) gives \(x = \frac{3}{2}\).

Step 4: Determine the Multiplicities

The multiplicity of a zero is the number of times the corresponding factor appears in the factored form.- The zero \(x = 0\) comes from \(x^3\), which has multiplicity 3.- The zero \(x = \frac{3}{2}\) comes from \((2x-3)^2\), which has multiplicity 2.

Key Concepts

Zeros of Polynomials

When we talk about the zeros of a polynomial, we are referring to the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation formed when you set the polynomial equal to zero. Finding these zeros is crucial because they provide key insights into the behavior of the polynomial function. To find the zeros, we factor the polynomial and then solve for the variable when each factor is set to zero.

• A polynomial of degree \(n\) can have up to \(n\) zeros, taking into account their multiplicities.
• Zeros can be real or complex numbers, though in real-world problems, we often focus on real zeros.

Understanding zeros helps in graphing polynomial functions and predicting the intersections of the function with the x-axis.

Factoring Polynomials

Factoring polynomials is a method used to express a polynomial as a product of its linear or simpler polynomial factors. This process makes it easier to find the zeros and analyze the polynomial's behavior.
When factoring polynomials, start by looking for the greatest common factor (GCF) of all terms and factor it out. For the polynomial \(f(x)=4x^5-12x^4+9x^3\), the GCF is \(x^3\). After factoring out \(x^3\), we write the remaining polynomial as \(x^3(4x^2 - 12x + 9)\).

• Next, factor the resulting polynomial further if possible, as was done with \(4x^2 - 12x + 9\) to get \((2x-3)^2\).
• Factoring completely is critical for finding all the zeros and understanding the graph of the polynomial.

Each polynomial factor corresponds to a potential zero of the function, derived by setting the factor equal to zero.

Multiplicity of Zeros

The multiplicity of a zero refers to how many times a particular zero appears for a polynomial function. It is essentially the exponent of the factor associated with that zero in the factored form of the polynomial.
A higher multiplicity affects the graph of the polynomial at the zero:

• If a zero's multiplicity is odd, the graph crosses the x-axis at that zero.
• If it is even, the graph touches the x-axis but does not cross it at that zero.

For the exercise polynomial \(f(x) = x^3 (2x - 3)^2\):
- The zero \(x = 0\), coming from the factor \(x^3\), has a multiplicity of 3, indicating that the graph will touch and cross the x-axis at \(x=0\) with a flattening.- The zero \(x = \frac{3}{2}\), coming from \((2x-3)^2\), has a multiplicity of 2, meaning the graph will touch but not cross the x-axis at \(x=\frac{3}{2}\). These properties help in visualizing the curve and predicting its behavior at various points.