Problem 47 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 2 x^{5}-8 x^{2}+9 x+4=12 x^{4}+3 x^{3}+4 x^{2}+1 $$

Step-by-Step Solution

Answer

Answer: The given equation is a quintic equation.

Step 1: Simplify the equation

In order to determine the largest exponent of x, we'll first simplify the equation by moving all terms to one side. $$ 2 x^{5} - 8 x^{2} + 9 x + 4 - 12 x^{4} - 3 x^{3} - 4 x^{2} - 1 = 0 $$

Step 2: Combine like terms

Now, we'll combine the like terms (terms with the same exponents) to simplify the equation further: $$ 2 x^{5} - 12 x^{4} - 3 x^{3} - 8 x^{2} - 4 x^{2} + 9 x + 4 - 1 = 0 $$ $$ 2 x^{5} - 12 x^{4} - 3 x^{3} - 12 x^{2} + 9 x + 3 = 0 $$

Step 3: Determine the degree

Now that the equation is simplified, we can determine its degree. The degree of a polynomial is the largest exponent of x in the equation. In this case, the largest exponent of x is 5 (from the term \(2x^5\)). Therefore, the degree of this equation is 5.

Step 4: Classify the equation

Since the degree of the equation is 5, it cannot be classified as linear (degree 1), quadratic (degree 2), or cubic (degree 3). The equation is a polynomial of degree 5, which can be called a quintic equation.

Key Concepts

Degree of a Polynomial

In algebra, the degree of a polynomial is a key concept that helps us understand its complexity and behavior. The degree of a polynomial is the highest power of the variable in the polynomial's expression.
For example, in the equation \(2x^5 - 12x^4 - 3x^3 - 12x^2 + 9x + 3 = 0\), the degree is 5 because the term \(2x^5\) has the largest exponent, which is 5.
This concept is used to classify polynomials:

• Linear: Degree 1, e.g., \(3x + 2\)
• Quadratic: Degree 2, e.g., \(x^2 + 3x + 2\)
• Cubic: Degree 3, e.g., \(2x^3 - 5x + 4\)
• Quartic: Degree 4
• Quintic: Degree 5

Recognizing the degree of a polynomial helps solve problems, as it indicates the number of solutions you might expect and the shape of its graph.

Quintic Equation

A quintic equation is a polynomial equation of degree 5. This means the highest power of the variable (usually \(x\)) is 5. Such equations resemble \(ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\). Quintic equations can be quite complex.
They tend to have up to five solutions, which could be real or complex numbers.
Unlike quadratic equations, quintic equations do not have a general algebraic solution.Previously solved approaches, like the quadratic formula for degree 2 or specific methods for cubic equations, don't apply generally to quintics. Mathematicians have developed numerous numerical methods to find solutions.Quintic polynomials are frequently found in advanced calculus and applied mathematics, particularly when modeling complex systems. Their solutions and behavior can be deeply insightful for understanding more intricate mathematical or scientific phenomena.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves combining like terms and making the expression as concise as possible. In the given example, \(2x^5 - 8x^2 + 9x + 4 - 12x^4 - 3x^3 - 4x^2 - 1 = 0\), simplifying was crucial.
Here are the steps taken:

• Move terms: Bring all terms to one side of the equation.
• Combine like terms: Add or subtract terms that have the same variable raised to the same power.
• Simplified form: Obtain \(2x^5 - 12x^4 - 3x^3 - 12x^2 + 9x + 3 = 0\).

Combining like terms avoids redundancy and errors, making it easier to solve or further manipulate the equation. Effective simplification reveals the true structure and degree of the polynomial.