Problem 47 - Beginning and Intermediate Algebra

Question

Factor completely. $$169-a^{2}$$

Step-by-Step Solution

Answer

The completely factored expression of \(169 - a^2\) is \((13 + a)(13 - a)\).

Step 1: Identify the difference of squares pattern

Rewrite the expression to better see the difference of squares pattern. $$169 - a^2 = (13^2) - a^2$$ Here, the expression is in the form of \(a^2 - b^2\), where \(a = 13\) and \(b = a\).

Step 2: Apply the difference of squares formula

Now, apply the difference of squares formula: \((a + b)(a - b)\) Substitute \(a = 13\) and \(b = a\) into the formula: \((13 + a)(13 - a)\)

Step 3: Write the final solution

The completely factored expression is: $$(13 + a)(13 - a)$$

Key Concepts

Difference of Squares

Understanding the difference of squares is crucial for mastering the factorization of certain algebraic expressions. The difference of squares is a special type of polynomial that takes the form: \[a^2 - b^2\]This expression can always be factored into the product of two binomials:\[(a + b)(a - b)\]It's called the 'difference' of squares because it subtracts one square from another.### Key Characteristics of Difference of Squares - Both terms are perfect squares.- There is a subtraction sign between the two terms.- It can be quickly identified and factored using its special formula.For example, in the expression \(169 - a^2\), we see it fits into this pattern.- Here, \(169\) is the same as \(13^2\), making it a perfect square.- \(a^2\) is obviously a perfect square.- They are connected by a subtraction sign, confirming the difference of squares structure.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs. They are the basic building blocks in algebra, allowing us to form equations and inequalities.### Understanding Algebraic Expressions- **Variables**: These are symbols (such as \(a\), \(b\), \(x\), \(y\)) that can represent unknown values or variables.- **Constants**: These are fixed numerical values, like \(169\) in our example.- **Operations**: Symbols like addition (+), subtraction (−), multiplication (×), and division (÷) are operations used within expressions.An example of an algebraic expression is \(3x + 2\). Here, \(3x\) consists of a variable \(x\) and a constant multiplier \(3\), while \(2\) is another constant. Expressions like \(169 - a^2\) utilize these components to create more complex forms that we can manipulate and solve.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into the simplest pieces, or factors, that when multiplied back together yield the original polynomial.### Principles of Polynomial Factorization- **Identify Patterns**: Recognizing structures like the difference of squares helps in quickly factoring polynomials.- **Simplify**: Breaking down each term into its simplest form aids in the factorization process.For instance, to factor \(169 - a^2\), first recognize it as a difference of squares:- Rewrite \(169\) as \(13^2\) so the expression becomes \(13^2 - a^2\).- Apply the difference of squares formula to factor it into \((13 + a)(13 - a)\).Factorization simplifies complex polynomials, making them easier to work with and solve. It has numerous applications, from solving quadratic equations to simplifying algebraic fractions.