Problem 19 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(x+6)(x-6)$$

Step-by-Step Solution

Answer

The product is \(x^2 - 36\).

Step 1: Recognize the Special Pattern

The given expression \((x+6)(x-6)\) can be recognized as a difference of squares pattern. The general form for the difference of squares is \((a+b)(a-b) = a^2-b^2\). Here, the binomials \((x+6)\) and \((x-6)\) match this pattern with \(a = x\) and \(b = 6\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula \((a+b)(a-b) = a^2-b^2\), substitute \(a = x\) and \(b = 6\) into the formula. This gives \(a^2 = x^2\) and \(b^2 = 6^2 = 36\).

Step 3: Perform the Calculation

Substitute the values from Step 2 into the formula to calculate the product: \(x^2 - 36\).

Step 4: Verify the Solution

Check the solution by expanding \((x+6)(x-6)\) directly: \((x+6)(x-6) = x^2 - 6x + 6x - 36 = x^2 - 36\). The terms \(-6x \) and \(+6x\) cancel out, confirming that the product is indeed \(x^2 - 36\).

Key Concepts

Binomial Multiplication

When dealing with binomial multiplication, it refers to multiplying two expressions, each consisting of two terms. A basic understanding of this process, often represented as

• FOIL (First, Outer, Inner, Last) method,
• using distributive properties,
• identifying patterns like the difference of squares,

is invaluable for solving polynomial equations. With binomials, we typically calculate each pair of terms, combine like terms, and simplify. The multiplication of \((x+6)(x-6)\) is made simpler by recognizing it fits a pattern known as the difference of squares which reduces calculation time.

Remember, practice teaches mastery, and observing the pattern can speed problem-solving.

Algebraic Identities

Algebraic identities are essential patterns or formulas that help streamline solving polynomial equations. These identities provide quick insights and shortcuts for complex algebraic multiplication.

In the problem \((x+6)(x-6)\), the difference of squares pattern is crucial: \((a+b)(a-b) = a^2 - b^2\). Here, \(a = x\) and \(b = 6\), transforming the calculation into \(x^2 - 36\). Recognizing these identities means solving these expressions is quicker and involves fewer computational steps.

Polynomial Operations

Polynomial operations encompass a variety of tasks including addition, subtraction, multiplication, and division of polynomial expressions. Having a grasp of basic operations is important for solving more complex expressions.

For binomials like \((x+6)(x-6)\), multiplication is the operation needed. Polynomial multiplication extends beyond basic numbers, utilizing distributive law and other algebraic properties to break down expressions into more manageable parts. It's about combining similar terms and simplifying expressions.

• First, identify the different elements,
• then execute each part of the operation,

until you reach a simplified polynomial result.

Multiplication Shortcuts

Multiplication shortcuts make complex algebraic expressions easier to solve. Especially in polynomial multiplication, like \((x+6)(x-6)\), identifying patterns help skip several steps. The difference of squares is one such shortcut, turning potential lengthy calculations into a straightforward process.

This shortcut allows us to deduce that \((a+b)(a-b) = a^2 - b^2\), giving us a quick solution of \(x^2 - 36\). Recognizing these shortcuts is beneficial, making algebra not only quicker but allowing for fewer errors in calculations. It's amazing what recognizing a pattern can do for simplifying complex tasks!

Developing an eye for these can dramatically improve both speed and accuracy in solving mathematical problems.