Problem 47 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 25-a^{2} $$

Step-by-Step Solution

Answer

Question: Factor the given binomial: 25 - a^2 Answer: (5 + a)(5 - a)

Step 1: Identify the square roots of the terms

Notice that the binomial is given by: $$ 25 - a^2 $$ Both terms are perfect squares. In particular, the square roots are: $$ \sqrt{25} = 5 \text{ and } \sqrt{a^2} = a $$

Step 2: Use the difference of squares formula

Utilize the difference of squares formula which states that for any two terms A and B: $$ A^2 - B^2 = (A + B)(A - B) $$ Here, A = 5 and B = a. Plug these values into the formula and factor the binomial:

Step 3: Factor the binomial

Using the difference of squares formula with A = 5 and B = a: $$ (5 + a)(5 - a) $$ This is the factored form of the given binomial.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic concept used in factoring certain types of binomials. This method helps us break down expressions like \(a^2 - b^2\) into a product of two binomials. What makes this technique powerful is its simplicity. The formula is: \[A^2 - B^2 = (A + B)(A - B)\] Here, \(A\) and \(B\) represent two distinct terms. This formula shows that an expression defined as a difference of squares can always be represented as a product of their sum and difference. For example, consider \(25 - a^2\) from the exercise. Notice it fits the pattern because both 25 and \(a^2\) are perfect squares. The square root of 25 is 5, and the square root of \(a^2\) is \(a\). Applying the formula, we express it as: \[(5 + a)(5 - a)\] This process of using the difference of squares effectively simplifies the binomial into two manageable terms.

Perfect Squares

Perfect squares are numbers or expressions that can be expressed as the product of an integer by itself. In algebra, they are crucial for recognizing patterns in polynomial expressions. They often appear as the squared terms in expressions suitable for the difference of squares factoring.

• A number like 25 is a perfect square because \(5 \times 5 = 25\).
• The expression \(a^2\) is also a perfect square because it is \(a \times a\).

When factoring, recognizing perfect squares allows you to use the difference of squares. This is exactly what makes an expression like \(25 - a^2\) easy to factor. Each term is a perfect square, empowering us to utilize specific algebraic identities, such as the difference of squares, to reduce the expression to simpler multipliers.

Binomials

Binomials are algebraic expressions containing two distinct terms. They are the foundation of many algebraic problems and patterns and can appear in various forms, such as \(x + y\), \(a - b\), or \(m^2 - n^2\). Knowing how to handle binomials is essential for simplifying and solving equations.In the context of the exercise \(25 - a^2\), the expression is a binomial. Here, it consists of two terms: the constant 25 and the variable expression \(a^2\). These terms, once identified, can often be manipulated using techniques like factoring. In this case, the binomial is suitable for the difference of squares method because both terms are perfect squares. By recognizing the pattern, you can employ the difference of squares formula to factor it as \((5 + a)(5 - a)\).Understanding how to factor binomials is crucial for simplifying algebraic expressions and finding solutions, laying the groundwork for tackling more advanced algebraic concepts.