Problem 112 - Introductory Algebra for College Students

Question

Factor by grouping: \(8 x^{2}-2 x-20 x+5\)

Step-by-Step Solution

Answer

The factored form of the given polynomial \(8 x^{2}-2 x-20 x+5\) by grouping is \((4x-1)(2x-5)\)

Step 1: Group the Terms

The first step is to group the terms in the given polynomial. As per the exercise, it can be grouped as follows: \(8x^{2}-2x\) and \(-20x+5\). So the polynomial becomes: \((8x^{2}-2x)- (20x-5)\)

Step 2: Factor Out GCF from Each Group

Next is to factor out the greatest common factor (GCF) from each group. The GCF of \(8x^{2}-2x\) is \(2x\), and for \(-20x+5\) is \(-5\). So factor out \(2x\) from \(8x^{2}-2x\) and \(-5\) from \(-20x+5\) respectively: \(2x(4x-1)-5(4x-1)\)

Step 3: Factor by Grouping

Now that the results inside the parentheses are the same, factor by grouping: \((4x-1)(2x-5)\)

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is vital when simplifying algebraic expressions. The GCF is the highest number that divides exactly into two or more numbers. When factoring polynomials, identifying the GCF of the terms is a critical first step. In our example, from the terms \(8x^{2}-2x\), the GCF is \(2x\); it's the largest expression that divides both terms without a remainder. Likewise, for \( -20x+5 \), the GCF is \( -5 \). Spotting the GCF allows us to simplify expressions, reduce fractions, and solve equations efficiently. It's a foundational concept in algebra that aids students in understanding more advanced topics later on.

Polynomial Factoring

Polynomial factoring is a process used in algebra to break down a polynomial into simpler terms, or 'factors', that when multiplied together give you the original polynomial. The purpose of factoring is to find the roots of the polynomial or to simplify the expression for further operations. The exercise we see uses 'factoring by grouping', which is one method of factoring polynomials. It involves rearranging and grouping terms so that we can factor out common expressions. As demonstrated in the step-by-step solution, once we factored out the GCF from each group, we were able to group the like terms, \(4x-1\), and factor them in a way that is more manageable, resulting in the factors \(2x-5\) and \(4x-1\). This method of factoring is an essential tool for solving polynomial equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operators (such as addition or multiplication). The expression represents a particular number or quantity. In the given exercise, \(8x^{2}-2x-20x+5\) is an algebraic expression comprising terms that are combined using arithmetic operations. It is important to understand how to manipulate these expressions using algebraic rules in order to simplify or factor them. This is a crucial skill in introductory algebra that leads to solving equations and understanding functions.

Introductory Algebra

Introductory algebra is the first level where students encounter concepts beyond basic arithmetic. It involves dealing with variables, understanding expressions, and solving equations. Factoring is a key topic within this subject, as it helps to simplify and solve quadratic and higher degree polynomials. The exercise provided is a classic example used in an introductory algebra class to illustrate how to break down a more complex expression into its factors. It also shows that algebra is not just about finding an answer but understanding the relationships between numbers and how to manipulate them using established rules.