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Chapter 5: Problem 77

Is the sum of two binomials always a binomial? Explain.

Short Answer

Expert verified
No, the sum of two binomials is not always a binomial. It depends on the terms of the binomials being added.

Step by step solution

01

Understanding a binomial

A binomial is an algebraic expression of the form \(a + b\), where \(a\) and \(b\) are terms which could be either numbers, variables, or the product of both.
02

Add Two Binomials

When you add two binomials, \(a + b\) and \(c + d\), the result is obtained by adding each term individually. So, \(a + b + c + d\). This is called collecting like terms.
03

Determine the Result

Looking at the result from Step 2 (\(a + b + c + d\)), see whether it is a binomial. The answer is that the result of adding two binomials may not always be a binomial because you might not be able to combine all terms to get a two-term expression. It depends on the given binomials.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operators like addition, subtraction, multiplication, and division. They form the basis of algebra and can range from simple to highly complex.
  • A simple example of an algebraic expression is a single number or a variable, such as 5 or x.
  • More complex expressions might involve multiple terms combined together, such as 3x + 2y - 4.
Understanding algebraic expressions is crucial because they allow us to model real-world problems in a mathematical format. By manipulating expressions, you can solve for unknowns and simplify calculations.
Adding Binomials
Adding binomials involves combining two algebraic expressions, each having exactly two terms. For instance, given the binomials \(a + b\) and \(c + d\), the sum is simply the aggregate of individual terms: \(a + b + c + d\).
Here are the steps to ensure you're adding them correctly:
  • Write down both binomials, one beneath the other. Align terms of similar degree if possible,.
  • Add corresponding terms together. When terms are alike, they can often be simplified, further reducing the expression.
Remember, the resultant expression may not always be a binomial unless further simplification shows only two distinct terms remain.
Collecting Like Terms
Collecting like terms is the process of simplifying expressions by combining terms with the same variables raised to the same power.
  • For example, in the expression \(3x + 5x + 2y\), the terms \(3x\) and \(5x\) can be combined because they are like terms, resulting in \(8x + 2y\).
  • The goal is to reduce the expression to its simplest form by grouping similar terms.
By collecting like terms, you can more easily understand the structure of the expression and work with it more efficiently in mathematical operations.
Binomial Sum
The sum of two binomials isn't always another binomial. When combining binomials through addition, the result can be influenced by the specific terms present in each binomial.
Consider adding binomials \(2x + 3\) and \(x + 2\). Adding these yields an expression: \(2x + 3 + x + 2\). Combining like terms results in \(3x + 5\), which is still a binomial. However, if more terms appear after simplifying, the result may not remain a binomial due to extra parts.
Key points:
  • The binomial sum depends on like terms in the original expressions.
  • If all terms cannot be combined, the result is not a binomial.
  • Always look for opportunities to simplify and combine terms.
As such, while adding binomials, always analyze the resultant expression to understand its final classification.

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Most popular questions from this chapter

\(7 x^{4} y^{-1}\)

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Writing Write a paragraph that explains how the adage "You can't add apples and oranges" might relate to adding two polynomials. Include several examples to illustrate the applicability of this statement.
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