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Chapter 6: Problem 10

Tell whether each of the following is an expression or an equation. $$ x^{3}=x^{2}-x+3 $$

Short Answer

Expert verified
Equation

Step by step solution

01

Identify the given mathematical statement

Look at the given statement: \[ x^{3} = x^{2} - x + 3 \]
02

Define expression and equation

An expression is a combination of numbers, variables, and operators (like +, -, *, /) without an equality sign. An equation is a statement that two expressions are equal, indicated by the '=' sign.
03

Analyze the given statement

Determine if the given statement includes an equality sign. The statement \( x^{3} = x^{2} - x + 3 \) has an equality sign ('='), indicating it states that two expressions are equal.
04

Conclusion

Based on the presence of the equality sign, the given statement \( x^{3} = x^{2} - x + 3 \) is an equation.

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Key Concepts

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

Expressions
An expression in mathematics is a combination of numbers, variables, and operators (like +, -, *, and /). It forms a value but does not make any comparison or relation, such as equality, between quantities. For example, look at the following expressions:
\[2x + 3\]
\[x^2 - 4y + z\]
These are simply groups of terms and variables combined to represent a value. Moreover, expressions can be simplified or factored but don't include equality signs. This differentiation is crucial when distinguishing expressions from equations.
Equations
Equations, unlike expressions, make a comparison between two quantities. They state that two expressions are equal. This is marked by the inclusion of an equality sign ('='). For example:
\[x + 2 = 5\]
Here, the equation states that 'x + 2' is equal to 5. Equations can be solved, which means finding the value of the unknown variable(s) that make the equation true. Here's another example from our exercise:
\[x^3 = x^2 - x + 3\]
This is an equation because it clearly shows that the expression on the left-hand side is equal to the expression on the right-hand side.
Equality Sign
The equality sign ('=') is what separates expressions from equations. It indicates that the expressions on either side of it are equal in value. In the given exercise statement:
\[x^3 = x^2 - x + 3\]
the equality sign is present in the middle, showing that the value of the expression on the left-hand side (\(x^3\)) is equal to the value of the expression on the right-hand side (\(x^2 - x + 3\)). This visual cue helps you to quickly identify whether a mathematical statement is an expression or an equation.
To summarize:
  • Expressions combine terms and variables but have no '=' sign.
  • Equations declare two expressions are equal, marked by an '=' sign.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ (a+b)^{2}-9 $$

If \(P(x)=x^{4},\) use factoring to simplify $$ P(a+h)-P(a) $$

Factor: $$ a x^{2}+2 a x+3 a+x^{2}+2 x+3 $$

Write an equivalent expression by factoring out the smallest power of \(x\) in each of the following. $$ x^{3 / 4}+x^{1 / 2}-x^{1 / 4} $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}-16 $$
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