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Magazine Chapter 3: Problem 261
For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=4 x^{5}-12 x^{4}+9 x^{3} $$
Short Answer
Expert verified
Zeros are 0 (multiplicity 3) and \(\frac{3}{2}\) (multiplicity 2).
Step by step solution
01
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: \[ f(x) = x^3 (4x^2 - 12x + 9) \].
02
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.
03
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 \(x = \frac{3}{2}\).
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
- 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 \((2x-3)^2\).
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.
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.
