Practice Math Questions for Praxis: Algebraic Factoring

In the following practice questions, you go in one direction to find the full factorization of an expression, and then in the opposite direction to find the original expression, given the full factorization.

Practice questions

1. Which of the following is the full factorization of 50x³y² – 10xy³ + 40xy?

   A. 10xy(5x²y + y² + 10) B. 10xy(5x²y – y² + 4) C. 10xy(5x² – y² + 10) D. 5xy(10x²y – y² + 10) E. 10x(5x²y – y² + 10y)

2. 4w⁴x⁷y³(7wxy² – 4w⁵x³y²) is the full factorization of what expression?

Answers and explanations

1. The correct answer is Choice (B). The first step in algebraic factoring of multiple-term expressions is to look for a greatest common factor (GCF) and factor it out of the expression if there is one. In this case, you can factor 10xy out of every term because every coefficient is divisible by 10 and every term has x and y in it. The lowest exponent for x in the terms is 1, and the lowest exponent of y is 1 (the 1 is understood in both cases). The variables of the greatest common factor are therefore x and y, both understood to have an exponent of 1. The exponent on each variable in the GCF is always the lowest exponent of that variable in the multiple-term expression.

   Because 10xy is the greatest common factor, factor it out of the expression and put it next to parentheses; inside the parentheses, you need to write the number of times 10xy goes into each term. You can figure that out by writing the number of times 10 goes into the coefficient of each term; next to that number, represent each variable with an exponent that's the number that must be added to the exponent in the GCF to get the exponent in the given expression.

   There's also an easier route. You can find the product of each choice until you get a product of 50x³y² – 10xy³ + 40xy; however, the correct choice must have a greatest common factor of 1 for the expression in the parentheses. The only choice that meets those requirements is Choice (B). You can multiply each choice by using the distributive property:

   

   There's an even quicker way to eliminate incorrect answer choices: You can multiply the expression outside the parentheses by the last term and note that Choice (B) is the only one that gives you 40xy.

2. 28w⁵x⁸y⁵ – 16w⁹x¹⁰y5

   The question implies that the expression in question is a full factorization, so you don't need to make sure it is. To answer this question, multiply the outside term by the inside expression using the distributive property.