Problem 12 - Elementary and Intermediate Algebra

Question

Factor. \(16 x^{2}-81\)

Step-by-Step Solution

Answer

The factored form is \((4x + 9)(4x - 9)\).

Step 1: Identify the Expression as a Difference of Squares

Notice that the expression given is in the form of a difference, specifically, it can be rewritten as \(a^2 - b^2\). Here, \(16x^2\) is \((4x)^2\) and 81 is \((9)^2\). This identifies the expression \(16x^2 - 81\) as a difference of squares.

Step 2: Recall the Factoring Formula for Difference of Squares

The formula for factoring a difference of squares is \(a^2 - b^2 = (a + b)(a - b)\). We will use this formula to factor the expression.

Step 3: Apply the Formula

Using the formula from Step 2, replace \(a\) with \(4x\) and \(b\) with \(9\). Applying the formula gives us: \((4x + 9)(4x - 9)\).

Step 4: Verify the Factors

To ensure the correctness of our factorization, we can expand \((4x + 9)(4x - 9)\) to check that it equals the original expression. Expanding gives \((4x)(4x) - (9)(4x) + (9)(4x) - (9)(9)\), which simplifies to \(16x^2 - 81\) after combining like terms. This confirms our factorization is correct.

Key Concepts

Difference of Squares

The concept of the **difference of squares** is a beautiful part of algebra that simplifies expressions quickly. The difference of two squares refers to an expression that takes the form \(a^2 - b^2\). Here's how it works:

• An expression like this involves two perfect squares separated by a minus sign.
• For example, when we see \(16x^2 - 81\), we can identify it as a difference of squares.
• The numbers in this expression can be represented as \((4x)^2 - (9)^2\).

Using the difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\), allows us to break down those big numbers into simpler factors. This way, we turn difficult multiplications into easier ones.

Algebra

**Algebra** is like the language of mathematics that helps us understand and describe patterns. When dealing with expressions like \(16x^2 - 81\), it's crucial to use algebra to recognize the hidden structures. Here are some key points about algebra in this context:

• Algebra involves using letters, like \(x\), to stand for numbers, which lets us create general formulas.
• In solving expressions, we often look for patterns like the difference of squares to make calculations smoother.
• Understanding these techniques can make algebra not only simpler but more powerful for solving various problems.

By mastering algebra, you can approach problems logically, step-by-step, and enjoy the satisfaction of reaching correct answers.

Polynomial Expressions

**Polynomial expressions** are a type of mathematical expression made up of sums and differences of terms. Here’s the breakdown of what a polynomial is and how it works:

• Each term in a polynomial is a product of a number and a variable raised to a power. For example, in \(16x^2\), 16 is the coefficient, and \(x^2\) shows the variable \(x\) squared.
• Polynomials can be simple with just one term or very complex with many terms.

In the expression \(16x^2 - 81\), we encounter a polynomial involving two terms. Recognizing this as a difference of squares helps transform it into a multiplication of two binomials \((4x + 9)(4x - 9)\). Using the properties of polynomials, factoring simplifies these complex expressions, making complex equations more manageable.