Problem 71 - A Graphical Approach to College Algebra

Question

Factor by any method. $$x^{2}+x y-5 x-5 y$$

Step-by-Step Solution

Answer

Factor the expression as \((x + y)(x - 5)\).

Step 1: Group Terms

Group the terms in the polynomial based on a common factor: \[ (x^2 + xy) + (-5x - 5y) \]

Step 2: Factor Out Common Factors

Factor out the greatest common factor from each group. For the first group, the common factor is \(x\), and for the second group, the common factor is \(-5\).1. From \((x^2 + xy)\), factor out \(x\): \[ x(x + y) \]2. From \((-5x - 5y)\), factor out \(-5\): \[ -5(x + y) \]

Step 3: Factor by Grouping

Now we have:\[ x(x + y) - 5(x + y) \]Notice that \((x + y)\) is a common factor. Factor \((x + y)\) out of the expression:\[ (x + y)(x - 5) \]

Key Concepts

Polynomial equations

Polynomial equations involve expressions made up of variables raised to various powers, often accompanied by coefficients. They look something like this:

• First, the terms are arranged with variables and coefficients.
• Next, these terms include operations like addition (+) or subtraction (-).
• The highest power of the variable determines the degree of the polynomial.

In the case of the given exercise, the polynomial expression is: \[ x^{2} + xy - 5x - 5y \]This expression involves multiple variables, namely \(x\) and \(y\), and contains both a quadratic term \(x^2\) and several linear terms. Understanding polynomial equations means you can rewrite expressions in simpler, factored forms—which is a fundamental aspect of solving such equations.

Common factor

Identifying the common factor is a crucial skill when dealing with polynomials. A common factor is a number or variable that divides evenly into each term of an expression. This simplifies the expression when factored out. Steps for Finding a Common Factor:

• Examine each term of the polynomial.
• Find the greatest factor shared among them.
• For variables, choose the lowest power present in each term as the common factor.

In our example, the expression is grouped into two parts: \((x^2 + xy)\) and \((-5x - 5y)\). Within each group, identify and factor out the common factor, which in this case is \(x\) for the first group, and \(-5\) for the second. Doing this simplifies the polynomial step by step, leading us to a clearer factored form.

Grouping method

The grouping method is a technique used to facilitate the factorization of polynomials, especially when they contain four terms. This method involves:

• Dividing the polynomial into groups that can each be factored separately.
• Factoring out any common factor within each group.
• Looking for a common binomial factor between the groups.

Using our example, we first group the polynomial as follows:\[ (x^{2} + xy) + (-5x - 5y) \]Next, we factor each group: - The common factor in \((x^2 + xy)\) is \(x\), resulting in \(x(x + y)\).- In \((-5x - 5y)\), the common factor is \(-5\), which gives us \(-5(x + y)\).The prime goal is to arrange the polynomial so that after factoring, one common factor (here, \(x + y\)) becomes apparent within the different groups. This facilitates further simplification.

Greatest common factor

The greatest common factor (GCF) is the largest factor shared by all terms in a polynomial. Finding the GCF helps simplify expressions significantly, making them easier to handle and solve. Here's how you find the GCF:

• Identify all potential factors of each term in the polynomial.
• Look for the largest factor that appears in all terms.
• In the case of variables, use the lowest power present across all terms.

In our exercise, the GCF plays a role in simplifying each group before final factorization. For the polynomial \(x^2 + xy - 5x - 5y\), recognize- For the term set \((x^2 + xy)\), the GCF is \(x\).- For the set \((-5x - 5y)\), the GCF is \(-5\).Once the GCF is factored out from each group, we've reduced the polynomial to simpler terms, ultimately leading to the fully factored form: \[(x + y)(x - 5)\].Using the GCF efficiently breaks down complex polynomials into their basic components, making further algebraic manipulation straightforward.