Problem 46 - Elementary Algebra

Question

For the following problems, list, if any should appear, the common factors in the expressions. $$ 15 x^{2}-30 x y^{2} $$

Step-by-Step Solution

Answer

Answer: The common factors of the given expression are 15 and x. The factored expression is $$15x(x-2y^{2})$$.

Step 1: Determine the GCD of the numerical coefficients

First, we need to find the GCD of the numerical coefficients, which are 15 and -30 in this expression. The GCD of 15 and -30 is 15.

Step 2: Identify the highest power of common variables

Now, we need to determine the highest power of common variables. In this case, the common variable is x. The highest power of x in both terms is x (since the first term has x^2 and the second term has x).

Step 3: Factor the expression using common factors

After identifying the GCD (15) and the highest power of the common variable (x), we will now factor the given expression using these common factors: $$ 15x^{2}-30xy^{2} = 15x(x-2y^{2}) $$ So, the common factors of the given expression are: 15 and x. And the factored expression is: $$ 15x(x-2y^{2}) $$

Key Concepts

Greatest Common Divisor

Understanding the Greatest Common Divisor (GCD) is essential when working with numbers and algebraic expressions. The GCD is the largest positive integer that divides two or more integers without leaving a remainder. For instance, when you have numerical coefficients such as 15 and -30, you look for the biggest number that can divide both 15 and 30 evenly. Here, 15 is the GCD because both numbers are multiples of 15. Finding the GCD helps simplify fractions or factor out algebraic expressions, making them easier to handle.

When dealing with algebraic expressions, identifying the GCD of numerical coefficients is the first step for simplification. In practice, you can list the factors of each number and find the largest common one, or use algorithms such as Euclidean algorithm for larger numbers. The role of the GCD in algebra is akin to finding a shared foundation upon which you can construct a simplified form of an expression.

Variable Powers

Variable powers indicate how many times a variable is multiplied by itself. In algebra, when you see an expression such as \( x^2 \), it tells you that \( x \) is squared, or multiplied by itself once (since \( x \) is equal to \( x^1 \)). The power (also called the exponent) provides a compact way to represent repeated multiplication, which is especially useful in simplification and factoring.

When factoring expressions, it's important to identify the highest power of the common variables present in all terms. For instance, if one term contains \( x^2 \) and another contains \( x \), the highest common power of \( x \) is the first power, since \( x^2 \) includes \( x \) as a factor, but not vice versa. Recognizing the correct power to factor out is a key step towards simplifying and solving algebraic expressions.

Algebraic Factorization

Algebraic factorization is the process of breaking down expressions into simpler, multiply-linked factors. Imagine it as the reverse of multiplication in algebra. Instead of combining values, you are seeking to identify component parts. Just like identifying building blocks in a structure, factorization finds the 'blocks' that, when multiplied together, give back the original expression.

In the solution provided, after finding the GCD (15) and realizing the common variable power (\( x \)), we use these to factor the expression. It becomes \( 15x(x - 2y^2) \). This shows that the original expression \( 15x^2 - 30xy^2 \) is made by multiplying the common factors \( 15x \) with the binomial \( x - 2y^2 \). Factoring simplifies expressions and is crucial for solving equations, simplifying rational expressions, and integral calculus operations.

By mastering algebraic factorization, variables and constants can be manipulated with greater ease, paving the way for deeper understanding and problem-solving in algebra.