Problem 48 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 49-16 a^{2} $$

Step-by-Step Solution

Answer

Question: Factor the expression \(49-16a^2\). Answer: \((7 - 4a)(7 + 4a)\)

Step 1: Identify the difference of squares

In order to factor the expression, we need to recognize that it is a difference of two squares. Rewrite the expression as: \((7^2)-(4a)^2\) Step 2: Use the factoring formula

Step 2: Use the factoring formula

Now that we have identified the expression as a difference of two squares, we can use the formula \((A - B)(A + B)\) to factor it. Here, \(A = 7\) and \(B = 4a\). Step 3: Apply the formula

Step 3: Apply the formula

Substitute \(A = 7\) and \(B = 4a\) in the formula \((A - B)(A + B)\) to get the factors: \((7 - 4a)(7 + 4a)\) The factored expression is: \((7 - 4a)(7 + 4a)\).

Key Concepts

Binomials

A binomial is a type of polynomial that consists of exactly two terms. These terms can involve variables, constants, or both but are combined using addition or subtraction. For example, in our exercise, the expression given is a binomial: 49 - 16a². This consists of the constant term '49' and the variable term '16a²', connected by a subtraction sign.
This simplicity of having only two terms makes binomials an ideal candidate for factoring, especially when they form a special pattern such as a difference of squares.
When working with binomials, look for these patterns to make factoring easier:

Factoring

Factoring in algebra involves breaking down an expression into products of simpler expressions. In the given exercise, the technique we used is specific to a "difference of squares" scenario.
The difference of squares is a form of binomial factoring where an expression is written as the difference (or subtraction) of two perfect square terms. A perfect square term is a number that can be expressed as another integer squared, like 49 which is 7².
Steps to factor the difference of squares include:

• Identifying two perfect squares in the binomial, like 49 and 16a² in the original problem.
• Writing them as squared terms: (7²) - (4a)².
• Applying the difference of squares formula: (A - B)(A + B).

By doing these, we have successfully factored the expression into (7 - 4a)(7 + 4a). This process simplifies the expression and is a fundamental skill in algebra.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's a powerful tool, especially when solving equations or factoring expressions like binomials.
Throughout algebra, you will encounter various expressions and equations that can often be simplified by factoring. Factoring not only makes expressions simpler but also reveals solutions hidden within the equations.
In our specific exercise, algebra helps by providing a method (difference of squares) to factorize the binomial and simplify it. By understanding fundamental concepts such as factoring, you gain a sharper edge in tackling a multitude of algebraic problems.
Remember, whether you are dealing with linear equations or complex polynomials, the principles and patterns you learn in algebra serve as a vital foundation for more advanced mathematics. In the context of binomials, mastering these techniques enhances your problem-solving capabilities significantly.