Problem 54 Eloise thinks the sum \(5 x^{2}+... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 54

Eloise thinks the sum $5 x^{2}+3 x^{4}$ is 8$x^{6} .$ What is wrong with her reasoning?

Short Answer

Expert verified

Eloise incorrectly combined unlike terms; $5x^2 + 3x^4$ cannot be simplified to $8x^6$.

Step by step solution

Understand the Expression

Consider the given expression: $5x^2 + 3x^4$. This expression is a polynomial with two terms.

Identify Powers of x

Look at the exponents of each term. The first term has an exponent of 2 ($x^2$), and the second term has an exponent of 4 ($x^4$).

Review Addition of Polynomials

When adding polynomials, you only combine like terms. Like terms are terms that have the same exponent.

Determine Like Terms

In the expression $5x^2 + 3x^4$, the terms $5x^2$ and $3x^4$ are not like terms because they have different exponents (2 and 4).

Sum of Unlike Terms

Since $5x^2$ and $3x^4$ are not like terms, their sum cannot be simplified further. It cannot be combined into a single term such as $8x^6$.

Explain the Error

Eloise's mistake is assuming that terms with different exponents can be combined by adding their exponents, which is incorrect. The sum $5x^2 + 3x^4$ should remain as it is because the two terms cannot be combined.

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Key Concepts

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

Understanding Like Terms

Like terms are terms within a polynomial that share the same variable raised to the same power. For example:

In the terms $2x^3$ and $5x^3$, both terms have the variable $x$ raised to the power of 3. Therefore, they are like terms.
Terms like $4x^2$ and $7y^2$ are not like terms because they have different variables.
Terms like $3x^2$ and $2x^3$ are not like terms because the exponents are different.

Identifying like terms is crucial in simplifying polynomials because only like terms can be combined. If the terms have different exponents or variables, they cannot be combined. This was Eloise's mistake: she tried to combine terms with different exponents, resulting in an incorrect solution.

Working with Exponents

Exponents tell us how many times a number, called the base, is multiplied by itself. For example:

The expression $x^2$ means $x$ is multiplied by itself: $x \times x$.
In $x^4$, $x$ is used as a factor four times: $x \times x \times x \times x$.

When dealing with polynomials, understanding exponents helps you recognize whether terms are like terms. Remember, you cannot combine terms with different exponents by just adding them. In Eloise's case, she incorrectly assumed that $5x^2$ and $3x^4$ could be added to form $8x^6$, which is not possible because the exponents 2 and 4 are different. The correct sum remains $5x^2 + 3x^4$.

Simplifying Polynomials

Simplifying polynomials involves combining like terms to make the expression as simple as possible. To simplify an expression:

First, identify all the like terms.
Next, combine the like terms by adding or subtracting their coefficients while keeping the same variable and exponent.
For terms that are not like terms, leave them as they are.

For example, consider the polynomial $4x^2 + 5x^2 - 3x + 2x^4$. To simplify it:

Combine like terms: $4x^2 + 5x^2 = 9x^2$.
Since $-3x$ and $2x^4$ are not like terms with anything else in this expression, leave them as they are.

So, the simplified polynomial is $9x^2 - 3x + 2x^4$. Eloise's mistake was that she attempted to combine terms that are not like terms, resulting in an incorrect simplification.

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91影视

Eloise thinks the sum \(5 x^{2}+3 x^{4}\) is 8\(x^{6} .\) What is wrong with her reasoning?

Short Answer

Step by step solution

Understand the Expression

Identify Powers of x

Review Addition of Polynomials

Determine Like Terms

Sum of Unlike Terms

Explain the Error

Key Concepts

Understanding Like Terms

Working with Exponents

Simplifying Polynomials

One App. One Place for Learning.

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