Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ 15 x^{5}-20 x^{4}=6 x^{3}-8 x^{2} $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) (with multiplicity 2), and other possible solutions from the cubic polynomial.

Step 1: Simplify and Rearrange

Start by moving all terms to one side of the equation to form a polynomial equation: \[ 15x^5 - 20x^4 - 6x^3 + 8x^2 = 0 \].

Step 2: Factor out the Greatest Common Factor

Identify the greatest common factor in the equation, which is \(x^2\), and factor it out: \[ x^2(15x^3 - 20x^2 - 6x + 8) = 0 \].

Step 3: Solve for the First Factor

The first factor is \(x^2 = 0\), hence \(x = 0\) is a solution. Since it is a repeated factor, \(x = 0\) has a multiplicity of 2.

Step 4: Solve the Cubic Polynomial

Now, solve the cubic polynomial \(15x^3 - 20x^2 - 6x + 8 = 0\). This step may involve using methods such as synthetic division or factoring by grouping if possible.

Step 5: Check for Additional Integer or Rational Root

Check for roots of the cubic using trial and error or the rational root theorem. Testing, say \(x = 2\), if it makes the equation zero.

Step 6: Verify Solutions from the Cubic Equation

Assuming \(x = 2\) is a root (from test or given solution), divide the cubic polynomial by \(x-2\) to find the other roots.

Step 7: Final Solutions

Combine all the roots from the polynomial \(x^2(15x^3 - 20x^2 - 6x + 8) = 0\). The complete set of solutions are \(x = 0\) and the solutions from the cubic equation.

Key Concepts

Factoring

Factoring is the process of breaking down a complex expression into a product of simpler expressions. This is an essential technique in algebra, especially when dealing with polynomial equations. The goal is to express the polynomial as a product of its factors, making it easier to solve for the variable of interest.

In this particular exercise, factoring played a critical role right from the beginning. First, the polynomial equation was simplified and rearranged to bring all terms to one side: \[15x^5 - 20x^4 - 6x^3 + 8x^2 = 0\].

Once in this form, the next step was to identify and factor out the greatest common factor (GCF), which in this case was \(x^2\). By factoring \(x^2\) out of each term, the equation became more manageable: \[x^2(15x^3 - 20x^2 - 6x + 8) = 0\].

This factorization step revealed two elements of the equation that needed separate consideration: the factor \(x^2\) and the cubic polynomial \(15x^3 - 20x^2 - 6x + 8\). Understanding how to factor effectively is pivotal in solving polynomial equations and finding all possible solutions.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, which means the highest power of the variable is three. In this exercise, the cubic polynomial under consideration was \(15x^3 - 20x^2 - 6x + 8\).

When solving a cubic polynomial, one often looks for methods such as synthetic division, factoring by grouping, or using the Rational Root Theorem to find its roots. In this problem, identifying \(x = 2\) as a potential root was a crucial step. This proposed root needed verification using trial and error or specific theorem-guided testing.

Once a root is confirmed, the next step involves dividing the cubic polynomial by the factor associated with the root, such as \(x - 2\), to reduce the polynomial to a simpler form. This process assists in uncovering the other roots, which may not be immediately obvious. With cubic polynomials, there could be three real roots, or one real and a pair of complex conjugate roots, depending on the equation’s discriminant or specific characteristic.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that all terms in a polynomial share. Finding the GCF is often the first step in the factoring process because it simplifies the polynomial, making it easier to handle.

In the provided exercise, the GCF was \(x^2\). By factoring \(x^2\) out of the entire polynomial, we reduced the equation from a complex expression into a simpler one:\[15x^5 - 20x^4 - 6x^3 + 8x^2 = x^2(15x^3 - 20x^2 - 6x + 8)\].

This step is not just mechanical; it's insightful. By identifying and factoring out the GCF, you expose smaller, more manageable components of the equation. This simplification is vital in reducing the polynomial's degree, paving the way for further factoring and solution finding. It also makes the subsequent factoring steps more straightforward, eliminating immediate redundancies and focusing on solving the core polynomial structure.

Multiplicity of Roots

Multiplicity of roots refers to the number of times a particular solution appears for a polynomial equation. If a root is repeated, it is said to have multiplicity greater than one.

In this exercise, after factoring out the GCF \(x^2\), the factor \(x = 0\) appeared, which has multiplicity 2 because it's derived from \(x^2 = 0\). This means the root \(x = 0\) is repeated twice as a solution to the polynomial equation.

• Understand multiplicity by recognizing that if \(x - r\) is a factor that's repeated in the polynomial, then \(r\) is a root with multiplicity equal to the number of times \(x - r\) is a factor.
• Multiplicity affects the graph of the polynomial: simple roots cross the x-axis, while repeated roots 'touch' it at the root's value, indicating a flattening effect where multiple roots exist.

Addressing the multiplicity of roots is vital, as it influences the completeness and accuracy of the solutions derived from the polynomial equation.