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Chapter 5: Problem 16

Fill in the blanks. A binomial of the form _____ is called the sum of two squares.

Short Answer

Expert verified
A binomial of the form \(a^2 + b^2\) is called the sum of two squares.

Step by step solution

01

Identification of Binomial Sum of Two Squares

To find the term that completes the statement, it is necessary to know what the sum of two squares in algebraic terms refers to. In algebra, the sum of two squares could be any binomial of the form \(a^2 + b^2\). Here, \(a\) and \(b\) are any real numbers, variables or terms.

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Key Concepts

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

Binomial
Understanding binomials is crucial for solving algebraic expressions. A binomial is simply a polynomial with two terms. These terms are usually separated by a plus or minus sign. In the context of the given exercise, a sum of two squares is a specific type of binomial. Polynomials like binomials can be manipulated using algebraic operations such as addition, subtraction, multiplication, and even division. The sum of two squares is just one structure of a binomial which takes a specific form that we鈥檒l explore in more detail below.
Algebraic Terms
Algebraic terms are the building blocks of algebraic expressions. They can consist of numbers, variables, or the products of numbers and variables. An algebraic term could be as simple as a constant number like 5, or as complex as a string of multiple variables and coefficients, such as 3xy^2. In the sum of two squares, each square term is an algebraic term. Recognizing different algebraic terms and understanding how to work with them paves the way for grasping more complex algebraic processes, including factoring and expanding binomials.
Real Numbers
Real numbers encompass a broad set of numbers including integers, rational numbers, and irrational numbers. All measurable quantities in algebra are represented by real numbers. When you encounter a square term in a binomial, it is implied that the base of that term is a real number. Since the square of any real number is non-negative, the sum of two squares will always be non-negative as well. This is a key property to remember when it comes to understanding the nature of certain algebraic expressions.
Variables
Variables are symbols used to represent numbers in algebraic terms and can take on different values. Commonly, letters at the beginning of the alphabet like a, b, or c are used to denote known quantities, while letters at the end like x, y, or z represent unknown quantities that need to be solved for. In the context of the sum of two squares, the variables can be part of our square terms鈥攕uch as in the binomial a^2 + b^2鈥攚here a and b are placeholders for real numbers. Variables allow algebraic expressions to be generalized and solved in terms of those symbols, rather than specific numbers.

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