Problem 4 - College Algebra

Question

Factor out the greatest common factor. $$ 4 x^{2}-8 x $$

Step-by-Step Solution

Answer

The factored form of the given expression \(4x^2 - 8x\) is \(4x(x - 2)\).

Step 1 - Identify the greatest common factor

The greatest common factor is the highest number that divides evenly into each of the terms. Here, the GCF of \(4x^2\) and \(8x\) is \(4x\), because both terms are divisible by \(4x\).

Step 2 - Divide each term by the GCF

Divide each term in the polynomial by the GCF. So, \(4x^2 / 4x = x\) and \(8x / 4x = 2\). This gives the factored expression: \(4x(x - 2)\) after the division.

Step 3 - Check the solution

To validate the solution, distribute the factored GCF back into the parenthesis. \(4x * x = 4x^2\) and \(4x * -2 = -8x\), which yields the original polynomial expression. This confirms that the factoring out was done correctly.

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents. These elements are combined using addition, subtraction, and, in some cases, multiplication. A polynomial is often written in the form of a sum, like \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\),where \(a_n, a_{n-1},..., a_1, a_0\) are constants, also known as coefficients, and \(n\) represents the degree of the polynomial.
Polynomials can have one or more terms, and their classification depends on the number of terms they contain.

• Monomial: A polynomial with one term, such as \(4x^2\).
• Binomial: A polynomial with two terms, like \(4x^2 - 8x\).
• Trinomial: A polynomial with three terms.

In the polynomial \(4x^2 - 8x\), there are two terms: \(4x^2\) and \(8x\). Each term consists of a coefficient, such as 4 and 8, and variables raised to a power, like \(x^2\) and \(x\). Understanding how to work with these components is crucial when factoring polynomials.

The Process of Factoring

Factoring is the mathematical process of breaking down a polynomial into a product of simpler terms or factors. The factored form is often easier to work with or provides valuable insights into the properties of the polynomial.
When factoring polynomials, one of the first steps is to identify the greatest common factor (GCF). The GCF is the largest factor that all terms in the polynomial share. In the exercise, the polynomial \(4x^2 - 8x\) contains the terms \(4x^2\) and \(8x\), and the GCF is \(4x\). This is because both terms can equally be divided by \(4x\).

• To factor \(4x^2 - 8x\), you divide each term by the GCF. \(4x^2 / 4x\) results in \(x\), and \(8x / 4x\) results in \(2\).
• The factored form becomes \(4x(x - 2)\).

This step creates a simpler expression that retains the essence of the original polynomial, but is easier to handle in algebraic operations like solving equations.

Core Principles of Algebra Applied

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. Factoring polynomials, such as the given \(4x^2 - 8x\), is a fundamental skill in algebra.
In algebra, finding the greatest common factor is often the first step in simplifying expressions or solving equations. By expressing polynomials like the one in this exercise in their factored form, you can easily solve for unknown variables or understand their behavior. This is especially useful when:

• Solving quadratic equations.
• Finding the roots of the polynomial, where the equation equals zero.
• Understanding the graphing of polynomial functions.

After any factoring operation, it is important to verify your work. You do this by distributing the factors back across each term in the expression to ensure it returns to its original form. In this case, multiplying \(4x\) with both \(x\) and \(-2\) gives you back \(4x^2 - 8x\). This reaffirms that the factoring has been done correctly, following the algebraic principles.