Problem 40 - College Algebra

Question

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

Step-by-Step Solution

Answer

Zeros: 0 (multiplicity 5), 2 (multiplicity 2).

Step 1: Rewrite the Polynomial

Begin by rewriting the function as \[ f(x) = 2x^4 (x^3 - 4x^2 + 4x) \]Observe that there's a common factor inside the parentheses. Factor the expression inside the parentheses further.

Step 2: Factor Completely

Factor the cubic polynomial inside the parentheses:\[ x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) \]The quadratic \( x^2 - 4x + 4 \) can be factored into \((x-2)^2\), resulting in:\[ x(x-2)^2 \]So the function is now \[ f(x) = 2x^4 \times x \times (x-2)^2\] which simplifies to \[ f(x) = 2x^5(x-2)^2 \].

Step 3: Find the Zeros

Set the function equal to zero and solve for \(x\): \[ 2x^5(x-2)^2 = 0 \].This equation gives us two factors to solve: 1. \( 2x^5 = 0 \)2. \( (x-2)^2 = 0 \).

Step 4: Solve for Zeros

Solve each factor separately:1. \( 2x^5 = 0 \) implies \( x = 0 \).2. \( (x-2)^2 = 0 \) implies \( x = 2 \).Thus, the zeros are \( x = 0 \) and \( x = 2 \).

Step 5: Determine Multiplicity

The multiplicity of a zero is determined by the exponent of its corresponding factor:- For \( x = 0 \), the factor is \( x^5 \) so the multiplicity is 5.- For \( x = 2 \), the factor is \( (x-2)^2 \) so the multiplicity is 2.

Key Concepts

Zero Multiplicity

In mathematics, the zero of a polynomial is the value of \( x \) that makes the entire polynomial equal to zero. Each zero can also have something known as a multiplicity, which tells us how many times that particular zero is repeated in the polynomial. Consider the polynomial from the exercise, which is \( f(x) = 2x^5(x-2)^2 \). When we factorize the expression, the terms \( x^5 \) and \( (x-2)^2 \) reveal the zeros of the polynomial.

• For the zero at \( x = 0 \), the factor \( x^5 \) indicates that this zero occurs 5 times because the exponent is 5. Therefore, the multiplicity of \( x = 0 \) is 5.
• For the zero at \( x = 2 \), the factor \( (x-2)^2 \) indicates that this zero occurs 2 times because the exponent here is 2. Thus, the multiplicity of \( x = 2 \) is 2.

Understanding zero multiplicity is crucial as it not only identifies the zeros but also provides insight into the nature of the polynomial graph. For example, a higher multiplicity means the graph will touch the x-axis at that zero and turn back, rather than simply crossing it.

Factoring Polynomials

Factoring polynomials is a fundamental process in algebra that involves breaking down a polynomial into simpler polynomials that can be multiplied together to obtain the original polynomial. In simpler terms, it's about finding the building blocks of a polynomial. In the exercise, the polynomial \( f(x) = 2x^4(x^3 - 4x^2 + 4x) \) is factored to make it easier to find its zeros.

Let's take a closer look at the factoring process:

• Firstly, you observe the polynomial \( x^3 - 4x^2 + 4x \) inside the parentheses and note common factors. Here, \( x \) is a common factor, which can be factored out, resulting in \( x(x^2 - 4x + 4) \).
• Secondly, the quadratic \( x^2 - 4x + 4 \) can be further factored into \((x-2)^2\).

By factoring completely, the polynomial simplifies to \( 2x^5(x-2)^2 \), revealing the zeros and their multiplicities. Factoring plays a crucial role in making polynomials manageable and understandable.

Cubic Polynomials

Cubic polynomials are those of degree three, which means they include terms up to \( x^3 \). They often take the form \( ax^3 + bx^2 + cx + d \). In our exercise, the function includes a cubic polynomial, \( x^3 - 4x^2 + 4x \), which we needed to simplify further.When dealing with cubic polynomials:

• Identify any common factor in all terms. Here, the common factor was \( x \), leading to \( x(x^2 - 4x + 4) \).
• Look out for patterns or formulas that make further factoring feasible, such as grouping terms or identifying perfect squares.

The simplification of a cubic polynomial often requires trial and error or the use of established factoring techniques. These techniques help to make the polynomial easier to manage, especially when aiming to find zeros.

Quadratic Factoring

Quadratic factoring involves breaking down a polynomial expression of degree two into a product of simpler expressions. Quadratics generally have the form \( ax^2 + bx + c \). In the provided solution, we encountered a quadratic within the cubic polynomial: \( x^2 - 4x + 4 \).Here's how quadratic factoring works:

• Look for patterns, such as perfect square trinomials, which can be rewritten as squares of binomials, like \((x-2)^2\) in this case.
• Apply known formulas or techniques, such as factoring by grouping or using the quadratic formula when needed.

In this exercise, recognizing \( x^2 - 4x + 4 \) as a perfect square trinomial allowed us to simplify the cubic polynomial quickly. Quadratic factoring is an essential tool for solving higher-degree polynomial equations effectively.