Problem 1 - Algebra and Trigonometry

Question

Consider the polynomial \(2 x^{5}+6 x^{4}+4 x^{3}\) How many terms does this polynomial have? List the terms: What factor is common to each term? Factor the polynomial: \(2 x^{5}+6 x^{4}+4 x^{3}=\)

Step-by-Step Solution

Answer

The polynomial has 3 terms: \(2x^5\), \(6x^4\), \(4x^3\). The common factor is \(2x^3\). The factored form is \(2x^3(x^2 + 3x + 2)\).

Step 1: Understanding the Problem

We are given the polynomial expression \(2 x^5 + 6 x^4 + 4 x^3\). We need to determine the number of terms, list these terms, identify a common factor among all the terms, and factor the polynomial.

Step 2: Counting and Listing Terms

The polynomial \(2 x^5 + 6 x^4 + 4 x^3\) consists of three distinct terms: \(2 x^5\), \(6 x^4\), and \(4 x^3\).

Step 3: Identifying the Common Factor

To find the common factor, identify the greatest common divisor (GCD) of the coefficients \(2\), \(6\), and \(4\), which is \(2\), and the lowest power of \(x\), which is \(x^3\). Therefore, the common factor is \(2x^3\).

Step 4: Factoring the Polynomial

Factor out the common factor \(2x^3\) from each term of the polynomial:\[2 x^5 + 6 x^4 + 4 x^3 = 2x^3(x^2 + 3x + 2)\].The expression inside the parentheses, \(x^2 + 3x + 2\), is the simplified polynomial after factoring out \(2x^3\).

Key Concepts

Greatest Common Divisor

The concept of the greatest common divisor (GCD) is crucial in simplifying polynomial expressions. It refers to the largest integer or polynomial that divides each term in the expression without leaving a remainder. When dealing with polynomials, we consider both numerical coefficients and variable parts. For example, to determine the GCD of the coefficients in our polynomial exercise, we looked at the terms: 2, 6, and 4.

• The GCD of these coefficients is 2, as it is the largest number that can evenly divide all three coefficients.

Additionally, for the variable parts, we find the lowest exponent across the terms for the variable "x". Here, it is the power 3 from the term with the smallest exponent for "x", making the lowest power of 'x' be \(x^3\). Ultimately, the common factor of the polynomial \(2x^5 + 6x^4 + 4x^3\) becomes \(2x^3\), forming an integral part of the factoring process.

Polynomial Terms

A polynomial is made up of terms, which can be thought of as building blocks of the expression. Each term in a polynomial is a product of a number (the coefficient) and a variable raised to a power (the exponent).For our polynomial \(2x^5 + 6x^4 + 4x^3\), it consists of three terms which can be individually identified as:

• \(2x^5\)
• \(6x^4\)
• \(4x^3\)

In this instance:- \(2x^5\) is the first term, where 2 is the coefficient and 5 is the exponent of \(x\). - \(6x^4\) is the second term, where 6 acts as the coefficient and 4 as the exponent.- \(4x^3\) finalizes the list, with 4 as its coefficient and 3 as the exponent.Recognizing these terms helps in identifying patterns, simplifying expressions, and factoring polynomials.

Common Factor

In algebra, identifying a common factor among terms in an expression is a critical step for simplification. A common factor is a number or expression that evenly divides each term without resulting in a remainder. For the polynomial \(2x^5 + 6x^4 + 4x^3\):

• We determine the common factor first by looking at the coefficients 2, 6, and 4. As noted before, the greatest common divisor here is 2.
• Then, we consider the variable parts; \(x^5, x^4, x^3\). The smallest exponent present in all the terms is 3, making the common power of \(x\) be \(x^3\).

Thus, the common factor of this polynomial is \(2x^3\). This means each term can be evenly divided by \(2x^3\), and it can be factored out of the polynomial to simplify it.

Algebraic Expression

An algebraic expression represents a mix of numbers, variables, and operations (addition, subtraction, multiplication, division) but doesn't include an equals sign, differentiating it from an equation. Polynomials are a specific type of algebraic expression characterized by terms consisting of variables raised to whole number exponents and corresponding coefficients.Considering the polynomial at hand, \(2x^5 + 6x^4 + 4x^3\), it demonstrates properties typical of algebraic expressions:

• It is a combination of several terms joined by addition.
• Each term itself is an algebraic expression due to the presence of constants and variables.

The study of algebraic expressions is fundamental as it lays the foundation for solving equations, understanding functions, and performing calculus. By simplifying an expression through factoring, polynomial expressions become more manageable and easier to work with.