Problem 86 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(9+c)(9-c)$$

Step-by-Step Solution

Answer

The special product for \((9+c)(9-c)\) is \(81 - c^2\).

Step 1: Identify a and b

In this case, our two terms are a = 9 and b = c.

Step 2: Apply the difference of squares formula

The difference of squares formula is given by (a+b)(a-b) = a² - b². We will apply this formula to our terms a = 9 and b = c: \((9+c)(9-c) = 9^2 - c^2\)

Step 3: Calculate the result

Now we will calculate the result using the formula that we derived in step 2: \(9^2 - c^2 = 81 - c^2\) So, the special product for (9+c)(9-c) is \(81 - c^2\).

Key Concepts

Special Products

Special products are algebraic expressions that follow specific patterns and can be solved more easily using known formulas. These patterns allow us to simplify calculations without extensive multiplication processes. One very well-known special product is the "difference of squares" formula, which applies to expressions in the form of

• extit{(a+b)(a-b) = a² - b²}

This formula works because when terms

• a + b
• a - b

are multiplied, the middle terms cancel out, leaving us with only

• a²
• - b²

This makes finding the product much more straightforward. In the provided exercise, the expression

• (9+c)(9-c)

was identified as a difference of squares. By applying the formula, the multiplication simplifies instantly to

• 81 - c²

understanding special products is fundamental in algebra as they speed up solving certain polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division. They are fundamental components of algebra that help us represent real-world problems mathematically. An expression can be as simple as a number and letter combination like

• 9 + c

or more complex with multiple terms like

• 9² - c²

In the exercise given,

• (9+c)(9-c)

is the starting algebraic expression, and based on the concept of special products, you transform it to

• 81 - c²

Algebraic expressions allow us to manipulate symbols to find solutions and simplify problems. They are foundational for forming equations and inequalities that model various scenarios.

Polynomials

Polynomials are a type of algebraic expression that consists of variables, coefficients, and exponents. They contain terms that are connected through addition or subtraction. For example,

• 81 - c²

is a polynomial of degree 2 because the highest exponent of the variable 'c' is 2. This specific polynomial presented in the problem is called a "quadratic". The solution of the exercise tested this polynomial concept by recognizing that

• (9+c)(9-c)
• 9² - c²

formed a polynomial where the multiplication of the terms resulted in a binomial, which fits the quadratic category. Noticing these patterns in polynomials can help predict behaviors of graphs, factor expressions, and solve equations effortlessly.