Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=3 x^{4}+6 x^{3}+3 x^{2} $$

Step-by-Step Solution

Answer

Zeros: 0 (multiplicity 2), -1 (multiplicity 2).

Step 1: Factor Out the Greatest Common Factor

Examine the function \(f(x)=3x^4 + 6x^3 + 3x^2\). Each term has a common factor of \(3x^2\). Factor \(3x^2\) out of the expression:\[f(x) = 3x^2(x^2 + 2x + 1)\]

Step 2: Factor the Quadratic Expression

Now examine the quadratic expression \(x^2 + 2x + 1\). Recognize that this is a perfect square trinomial. It can be factored as:\[x^2 + 2x + 1 = (x + 1)^2\]Substitute this into the expression:\[f(x) = 3x^2(x + 1)^2\]

Step 3: Identify the Zeros

The function \(f(x) = 3x^2(x + 1)^2\) is set to zero to find the zeros:\[3x^2(x + 1)^2 = 0\]This equation is true if \(3x^2 = 0\) or \((x + 1)^2 = 0\).

Step 4: Solve for Zero from Each Factor

1. For \(3x^2 = 0\), divide both sides by 3 to get \(x^2 = 0\) which gives the solution \(x = 0\).2. For \((x + 1)^2 = 0\), take the square root of both sides to get \(x + 1 = 0\), leading to \(x = -1\).

Step 5: Determine the Multiplicity of Each Zero

The zero \(x = 0\) comes from the factor \(3x^2\), which implies it has multiplicity 2.The zero \(x = -1\) comes from the factor \((x + 1)^2\), which implies it has multiplicity 2.

Key Concepts

Multiplicity of Zeros

When we talk about polynomial zeros, we're speaking about the values of \(x\) that make the polynomial equal to zero. Each zero can also have something called a "multiplicity". This term just tells us how many times a particular zero occurs. Simply put, it's the number of times a zero is "repeated" in the factorization of the polynomial.

For example, in our polynomial \(f(x) = 3x^2(x + 1)^2\), when set to zero, we solve this to find the zeros as \(x = 0\) and \(x = -1\). But both of these zeros appear within factors that have exponents greater than one:

• The zero \(x = 0\) comes from the factor \(x^2\), indicating it has a multiplicity of 2.
• The zero \(x = -1\) comes from the factor \((x + 1)^2\), indicating it also has a multiplicity of 2.

Understanding multiplicity helps us know the behavior of the polynomial graph near these zeros. If a zero has an odd multiplicity, the graph passes through the x-axis at that point. If the multiplicity is even, the graph just touches the x-axis.

Factoring Polynomials

Factoring polynomials is like breaking down a larger number into its prime factors. It involves expressing a polynomial as a product of its simpler component polynomials. This process is crucial for solving equations, as it can reveal the roots (or zeros) quickly!

For the polynomial \(f(x) = 3x^4 + 6x^3 + 3x^2\), our first step involves finding the greatest common factor (GCF) of all the terms. We notice that all terms can be divided by \(3x^2\):

• This means \(3x^2\) is the GCF, and we can "factor out" this term.
• The expression simplifies to \(f(x) = 3x^2(x^2 + 2x + 1)\).

This step is essential as it simplifies the polynomial, allowing us to factor other parts more easily and eventually solve for the zeros.

Quadratic Expression

A quadratic expression is a type of polynomial that takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Our polynomial has a quadratic expression in it, which is \(x^2 + 2x + 1\).

Quadratic expressions can often be factored into two linear expressions, or recognized as a different type of special polynomial like a perfect square. Factoring these expressions is fundamental in finding the zeros, because once they're factored, setting each factor to zero gives us the solutions. In our example, the quadratic \(x^2 + 2x + 1\) is actually a perfect square, we'll discuss below.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression. It results from squaring a binomial. That sounds complex, but it really means something simple: if you take a binomial (like \((x + a)\)) and square it, you get a trinomial (like \((x^2 + 2ax + a^2)\)).

In our polynomial, we have \(x^2 + 2x + 1\), which is a perfect square trinomial because it equals \((x + 1)^2\):

• Here, \(a = 1\), so the expression is \(([x + 1])^2 = x^2 + 2x + 1\).

Recognizing and factoring perfect square trinomials is incredibly useful. It allows us to factor seemingly complex expressions quickly. Once factored, they immediately expose the zeros of the polynomial and their multiplicities due to the squared factor. Perfect square trinomials are quite common in polynomials, so spotting them early helps in simplification and solving equations.