In Exercises 15–58, find each product. (x−3)2

Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 45

Chapter 1, Problem 45

In Exercises 15–58, find each product. (x−3)²

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Recognize that the expression \((x - 3)^2\) represents a binomial squared. This means you will expand it using the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).

Identify the terms in the binomial \((x - 3)\): here, \(a = x\) and \(b = 3\).

Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\) to the given expression. Substitute \(a = x\) and \(b = 3\) into the formula.

Simplify each term: \(a^2 = x^2\), \(-2ab = -2(x)(3) = -6x\), and \(b^2 = 3^2 = 9\).

Combine the simplified terms to write the expanded form of the expression: \(x^2 - 6x + 9\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)^n. In this case, (x - 3)^2 can be expanded using the formula (a - b)^2 = a^2 - 2ab + b^2, where a = x and b = 3. Understanding this concept is essential for correctly applying the expansion to find the product.

Square of a Binomial

The square of a binomial is a specific case of binomial expansion where a binomial expression is multiplied by itself. For (x - 3)^2, this means multiplying (x - 3) by (x - 3). The result will yield a quadratic expression, which is a polynomial of degree two. Recognizing this pattern helps in simplifying and solving similar algebraic expressions.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the form ax^2 + bx + c, where a, b, and c are constants, and a is not zero. The result of expanding (x - 3)^2 will yield a quadratic expression. Understanding the structure of quadratic expressions is crucial for further analysis, such as factoring, graphing, or solving equations derived from them.