Problem 64 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$(x+5)^{2}-y^{2}$$

Step-by-Step Solution

Answer

\((x+5+y)(x+5-y)\)

Step 1: Identify the formula

We have been given the polynomial \((x+5)^2 - y^2\). This expression is in the form of a difference of squares, which is \(a^2 - b^2\).

Step 2: Recognize the components

Set \(a\) as \(x+5\) and \(b\) as \(y\) so that the expression fits into the formula \((a^2 - b^2 = (a+b)(a-b))\). Here, \(a = x+5\) and \(b = y\).

Step 3: Apply the difference of squares formula

Using the difference of squares formula \((a^2 - b^2 = (a+b)(a-b))\), substitute \(a\) and \(b\) with \(x+5\) and \(y\), respectively. Thus, the factorization is \(((x+5) + y)((x+5) - y)\).

Step 4: Write the final factored expression

The factored form of the given polynomial is \((x+5+y)(x+5-y)\). This expression is fully factored using integers.

Key Concepts

Difference of Squares

When dealing with polynomial expressions like \(x^2 - y^2\), the concept of the "difference of squares" is essential. This happens when you have two perfect squares subtracted from each other. The formula takes the form \(a^2 - b^2 = (a + b)(a - b)\), where \(a\) and \(b\) are any expressions or numbers. This formula helps to simplify expressions significantly by breaking them down into their multiplied components.
The expression \( (x+5)^2 - y^2 \) fits this category as it can be rewritten in the form of the difference of squares. Here, \( (x+5)\) takes the place of \(a\), and \(y\) is \(b\). By substituting these values into the difference of squares formula, you can factor it efficiently as \( (x+5+y)(x+5-y)\).
Implementing this algebraic identity simplifies the process of factoring polynomials and helps in solving complex algebra problems with ease. This approach is crucial when working on higher-level math problems that involve polynomial expressions.

Factoring Techniques

Factoring techniques are a set of methods used to simplify polynomials by expressing them as a product of simpler polynomials. Understanding these methods is key to solving polynomial equations effectively.
One vital technique is leveraging special formulas like the difference of squares. This is a powerful method as it allows you to factor expressions like \( (x+5)^2 - y^2 \) by recognizing it and applying the standard formula, which results in \( (x+5+y)(x+5-y)\).
Besides the difference of squares, other techniques include:

• Common factoring, where you take out the greatest common factor of a polynomial.
• Trinomial factoring, which applies to quadratic expressions of the form \(ax^2 + bx + c\).
• Grouping, used to factor polynomials with four or more terms.

Each of these methods requires recognizing patterns and understanding when and how to apply appropriate formulas. The practice of using the correct technique not only simplifies computations but also builds a strong foundation for more challenging algebraic tasks.

Polynomial Expressions

A polynomial is an expression consisting of variables and coefficients, displayed in terms of powers of variables. They might look complex, but breaking them down is easier once you understand basic operations like addition, subtraction, multiplication, and division as applied to polynomials.
The expression \( (x+5)^2 - y^2 \) is a polynomial because it involves two terms, each of which is a square, hence fitting into the moniker of a polynomial with distinct terms. Understanding polynomials is crucial because of the role they play in different areas of math, from algebra to calculus.
When handling polynomials, remember:

• Degrees: The degree of a polynomial is the highest power of the variable present.
• Terms: Polynomials are made up of terms separated by addition or subtraction.
• Standard form: Polynomials are usually written in descending order of the powers.

Mastering polynomial expressions, like learning how to factor them, is a critical skill in mathematics. It opens the doors to solving quadratic equations, modeling real-world situations, and interpreting mathematical data effectively.