Problem 33 - Elementary Algebra

Question

Factor completely. $$ -100 x y 3+4 x 3 y $$

Step-by-Step Solution

Answer

The completely factored expression is \(4xy(x - 5y)(x + 5y)\).

Step 1: Identify Common Factors

The expression given is \(-100xy^3 + 4x^3y\). First, we identify the greatest common factor (GCF) from both terms. Looking at the coefficients -100 and 4, the GCF is 4. The variables present are both \(x\) and \(y\), so the GCF including the variables is \(4xy\).

Step 2: Factor Out the GCF

Now, we factor the GCF \(4xy\) out of each term in the expression. The expression is rewritten as:\(4xy(-25y^2 + x^2)\).

Step 3: Check for Further Factoring

Examine the factored expression \(-25y^2 + x^2\) for further factoring. This expression is a difference of squares as it can be seen as \((x^2 - (5y)^2)\).

Step 4: Factor the Difference of Squares

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Applying this, we have:\((x - 5y)(x + 5y)\).

Step 5: Write the Fully Factored Expression

Combine all the factored parts to express the complete factorization:\(4xy(x - 5y)(x + 5y)\). This is the completely factored form of the original expression.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an important concept in factoring algebraic expressions. It refers to the largest factor shared by all terms in an expression. To find the GCF of a polynomial, you look at both the coefficients (numbers in front of variables) and the variables themselves.

Here's how to identify the GCF:

• Check the coefficients. For example, with the numbers -100 and 4, the GCF is 4.
• Look at the variables in each term. If both have variables such as \(x\) and \(y\), include them in your GCF as well.

Once you've identified the GCF, factor it out of each term. This means dividing each part of the expression by the GCF and rewriting the expression with the GCF outside of the parentheses. In our example, with a GCF of \(4xy\), you get \(4xy(-25y^2 + x^2)\). This simplifies the expression considerably by showing common factors upfront.

Difference of Squares

The difference of squares is a special algebraic pattern that helps in further breaking down expressions like \(-25y^2 + x^2\). It follows the formula \(a^2 - b^2 = (a - b)(a + b)\), where both \(a\) and \(b\) are squares of an expression.

This type of factoring is particularly useful when dealing with expressions that don't have a common factor other than 1. To use this technique:

• Identify expressions that neatly fit into the \(a^2 - b^2\) format.
• Recognize that each term is actually a squared value, for example, \(x^2\) and \((5y)^2\).

With \(-25y^2 + x^2\), change it into \((x - 5y)(x + 5y)\) using this formula. This method breaks the expression down into factors that can be easier to work with. Factoring through difference of squares can simplify complex algebraic expressions, making them more manageable and easier to solve.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition and subtraction. They are foundational in algebra, allowing us to create equations and solve problems. Understanding how to manipulate these expressions is key to mastering algebra.

Here are some basic features of algebraic expressions:

• They consist of terms, which can be constants, variables, or products of both.
• Terms are connected by addition or subtraction, not multiplication or division.

In our example, the original expression \(-100xy^3 + 4x^3y\) is an algebraic expression. Solving algebra problems often involves simplifying these expressions by factoring, combining like terms, or applying mathematical operations.

Mastering methods like finding the GCF, identifying difference of squares, and manipulating algebraic expressions allows for efficient problem solving in more complex equations. So, sharpening your understanding and skills in handling algebraic expressions is essential for any math journey.