Problem 18 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ y^{2}-49 $$

Step-by-Step Solution

Answer

Answer: The factored form of the given binomial is $$(y-7)(y+7)$$.

Step 1: Identify the values of 'a' and 'b'

The given binomial can be written as $$y^2 - 7^2$$. Here, $$a = y$$ and $$b = 7$$.

Step 2: Apply the difference of squares formula

Using the formula $$a^2 - b^2 = (a-b)(a+b)$$, apply the values of $$a$$ and $$b$$ to factor the given binomial.

Step 3: Factor the binomial

Replace the values of $$a$$ and $$b$$ in the formula: $$ y^2 - 49 = y^2 - 7^2 = (y-7)(y+7) $$ So the factored form of the given binomial is $$(y-7)(y+7)$$.

Key Concepts

Difference of Squares

Understanding the difference of squares concept is crucial in factoring binomials. This is a special case in algebra where a binomial is composed of two squared terms separated by a subtraction sign. The generic form of a difference of squares is \[a^2 - b^2\].

When factoring a difference of squares, the equation can be broken down into two binomials as follows: \[a^2 - b^2 = (a - b)(a + b)\]. Here, 'a' and 'b' represent any algebraic expression, including numbers, variables, or a combination of both.

Let's use this knowledge to refactor our original exercise, \[y^2 - 49\]. Recognizing that 49 is a perfect square (\[7^2\]), we identify our 'a' as \(y\) and our 'b' as \(7\). Applying the difference of squares formula, we factor the binomial to obtain \[ (y-7)(y+7)\], which represents two possibilities in which the difference of their squares equals the original binomial expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. These expressions can range from simple, such as \(5x + 3\), to complex, involving multiple terms and variables.

Factors of algebraic expressions are smaller expressions that, when multiplied together, yield the original expression. Identifying factors is a core skill in elementary algebra, and it often includes dealing with polynomials and binomials, such as the one in our exercise \[y^2 - 49\].

When faced with algebraic expressions like the one in our exercise, we apply specific factoring techniques, such as the difference of squares, to make the problem easier to handle and solve.

Elementary Algebra

Elementary algebra serves as the foundation of more advanced math courses. It entails operations with numbers and variables, and allows for the solving of equations and understanding of functions.

Key concepts within elementary algebra include understanding variables, coefficients, and the different properties of operations (e.g., associative, distributive). Additionally, the ability to manipulate and factor algebraic expressions to simplify or solve equations is an essential skill. In our example, factoring the binomial \[y^2 - 49\] requires knowledge of elementary algebra principles such as identifying square terms and performing operations with them.

Binomial Factorization

Binomial factorization is the process of breaking down a binomial expression, or a polynomial with two terms, into a product of simpler factors. It requires recognizing patterns and applying specific formulas or techniques for factoring.

For the exercise \[y^2 - 49\], the process involves identifying it as a classic difference of squares and then using the appropriate formula to factor it. It's essential to first recognize it as a difference of squares and then apply the formula \[a^2 - b^2 = (a - b)(a + b)\], as demonstrated in the steps provided.

Learning the binomial factorization technique can simplify solving equations, graphing functions, and can also play a role in calculus when finding limits. Therefore, mastering this elementary algebra concept is beneficial for future mathematical challenges.