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Chapter 4: Problem 49

Find each product. $$ (x+9)(x-9) $$

Short Answer

Expert verified
(x+9)(x-9) = x^2 - 81

Step by step solution

01

- Identify the expression

Recognize that the given expression (x+9)(x-9) is a difference of squares.
02

- Apply the Difference of Squares Formula

Recall the difference of squares formula: (a + b)(a - b) = a^2 - b^2 In this case, a = x and b = 9.
03

- Substitute the values into the formula

Substitute a and b into the difference of squares formula: (x+9)(x-9) = x^2 - 9^2
04

- Simplify the expression

Finally, simplify the expression: x^2 - 81

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
An algebraic expression is a mathematical phrase combining numbers, variables, and arithmetic operations. It does not include an equality sign, which distinguishes it from an equation. Let's break it down:
  • Numbers: These are constants like 1, 2, or 9.
  • Variables: These are symbols like x, y, or z representing unknown values.
  • Operations: These include addition, subtraction, multiplication, and division.
In the given exercise, the expression (x+9)(x-9) combines variables and numbers using multiplication. Recognizing different types of expressions, such as the difference of squares, helps simplify these expressions efficiently.
Factoring
Factoring is the process of breaking down a complex expression into simpler elements called factors. In algebra, it's like reversing the distributive property.

For example, (x^2 - 81) can be factored using the difference of squares rule. This rule states: (a^2 - b^2) = (a + b)(a - b)

So, if we have (x^2 - 9^2), it can be factored to (x + 9)(x - 9). This is exactly the principle used in the step-by-step solution of our original problem. Recognizing and applying factorizations simplifies solving complex algebraic equations.
Polynomials
Polynomials are expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication. They can have one or more terms. For example, (3x - 2) is a simple polynomial, while (x^3 + 2x^2 - 5x + 7) is more complex.

In our exercise, (x+9)(x-9) is a product of two binomials (polynomials with two terms). Using the difference of squares, we simplified it to (x^2 -81). This result, (x^2 - 81), is a polynomial known as a quadratic, as it is in the form ax^2+bx+c (where a, b, and c are constants). This shows how understanding polynomials helps in recognizing and solving algebraic expressions efficiently.

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