Problem 3 - Algebra 2

Question

Find the GCF of each expression. Then factor the expression. $$ x^{2}-2 x $$

Step-by-Step Solution

Answer

The GCF of the expression \(x^2-2x\) is \(x\), and the factored expression is \(x(x-2)\).

Step 1: Identify the greatest common factor (GCF)

Examine each term of the algebraic expression to determine the greatest common factor. In the expression \(x^2-2x\), both terms have \(x\) as a common factor. Since there are no other common numerical factors, the GCF is \(x\).

Step 2: Factor out the GCF

After identifying the GCF, factor it out of each term in the expression. This involves dividing each term by the GCF and using the result to write an equivalent factored expression. For \(x^2-2x\), divide each term by \(x\) to get \(x(x-2)\).

Key Concepts

Greatest Common Factor

Understanding the concept of the greatest common factor (GCF) is essential in algebra. It refers to the largest number that divides two or more numbers without leaving a remainder. To find the GCF in algebraic expressions, look for the highest exponent of common variables and the largest numerical value that evenly divides the coefficients of those variables.

For example, consider an expression like \(a^3b^2 - 4a^2b^3\). The GCF would be \(a^2b^2\) because it is the highest power of both \(a\) and \(b\) that can be found in each term of the expression. Identifying the GCF is the first critical step in simplifying and factoring algebraic expressions.

GCF in Algebra

When dealing with GCF in algebra, it's not just about numbers but also the variables and their exponents. The process of identifying the GCF among algebraic terms overlaps with our understanding of the GCF for numbers, as both involve division. However, instead of just looking at numbers, we also look at variables.

In the provided exercise, \(x^2-2x\), the GCF was determined to be \(x\). This is because \(x\) is the common variable in both terms and there are no numerical factors (other than 1) that are common. To factor algebraic expressions using the GCF, divide each term by the GCF and express the original expression as a product of the GCF and the resulting simplified terms.

Factoring Quadratics

Factoring quadratics is a process commonly used in solving quadratic equations. A quadratic expression takes the form of \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants with \(a eq 0\).

To factor a quadratic, one generally looks for two binomials that multiply together to give the original quadratic expression. This often involves finding two numbers that multiply to give \(ac\) and add to give \(b\). For more complex quadratics, where straightforward factoring isn't possible, methods such as completing the square or using the quadratic formula may be necessary. Factoring is not only a critical skill for solving equations, but it also plays a significant role in simplifying expressions and finding zeroes of functions.

Algebraic Factorization

Algebraic factorization is the process of breaking down complex algebraic expressions into simpler, multiply-linked factors. The key to mastering algebraic factorization is recognizing patterns, such as the difference of squares, perfect squares, and sum or difference of cubes.

Once you identify a pattern or determine the GCF, you can apply different factoring techniques. The goal of factorization is to make an expression easier to manipulate and solve. It's a powerful tool not just for solving equations, but for simplifying expressions, finding function intercepts, and performing operations with algebraic fractions. Equipping oneself with a solid understanding of various factoring techniques is invaluable for progressing in algebra and higher levels of mathematics.