Factoring Expressions to Obtain a Product of Sums: TW + UY + V

As an excellent high school mathematics teacher, I'm excited to share my knowledge on factoring expressions to obtain a product of sums in the form of TW + UY + V. This is a fundamental skill that students need to master in order to succeed in advanced algebra and beyond.

Understanding the Structure of the Expression

The expression we're going to factor is in the form of TW + UY + V, where T, U, W, and Y are variables, and V is a constant. To factor this expression, we need to identify the common factors among the terms and then use the distributive property to rewrite the expression as a product of sums.

Step 1: Identify the Common Factors

The first step in factoring the expression TW + UY + V is to identify the common factors among the terms. In this case, the common factor is the variable V. We can factor out V from the expression, leaving us with:

V(T + U)

Step 2: Factor the Remaining Expression

Now that we've factored out the common factor V, we need to factor the remaining expression (T + U). We can do this by recognizing that the expression is a sum of two terms, T and U.

To factor the expression (T + U), we can use the distributive property. This means that we can factor out the greatest common factor (GCF) from the two terms, and then write the remaining factors as a sum.

In this case, the GCF of T and U is 1, so we don't need to factor out any additional terms. The factored expression becomes:

V(T + U)

Final Factored Expression

Putting it all together, the factored expression in the form of TW + UY + V is:

V(T + U)

This factored expression is in the desired form of a product of sums, where the first factor is the common factor V, and the second factor is the sum of the variables T and U.

By understanding the structure of the expression and applying the steps of factoring, you can easily factor any expression in the form of TW + UY + V to obtain a product of sums. This is a valuable skill that will serve you well in your high school mathematics studies and beyond.