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Chapter 5: Problem 7

\(7 x-5 x^{2}+10\)

Short Answer

Expert verified
The given expression \(7 x-5 x^{2}+10\) when simplified becomes -5\(x^{2}\) + 7x + 10. No further simplification can be made.

Step by step solution

01

Identify all Terms

The first step is to identify all the terms of the expression. The expression given is a quadratic expression, which means it contains terms of \(x^{2}\), \(x\) and constants. In this case, the terms are -5\(x^{2}\), 7x and 10.
02

Rearrange Terms

The next step is to rearrange the terms of the expression in descending order of their powers. The terms should be listed starting with the highest power of \(x\). Our expression then becomes -5\(x^{2}\) + 7x + 10.
03

Consider Possible Simplification

Since no two terms in -5\(x^{2}\) + 7x + 10 are like terms or can be combined or reduced, the quadratic expression is already in its simplest form.

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Key Concepts

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

Identifying Terms in Quadratic Expressions
In algebra, understanding and identifying the terms of an expression is the first crucial step. Terms are the building blocks of an expression, and they consist of numbers and variables linked by multiplication or division. In the quadratic expression given here,
\(7x - 5x^2 + 10\),
we need to spot each distinct component.
  • -5\(x^2\): This is our quadratic term because it involves \(x\) squared.
  • 7x: This is the linear term where the variable is raised to the power of one.
  • 10: This is a constant term since it's just a number without any variables.
Identifying terms correctly forms the basis for further manipulation, making it easier to arrange or simplify the mathematical expression.
Rearranging Terms for Clarity
Once the terms are identified, rearranging them can help reveal the structure of a quadratic expression. A common convention is to list terms in descending order of their exponents. This order is not required, but it helps standardize expressions and can make working with them easier.
In our expression, we rearrange the terms based on the power of \(x\):
  • -5\(x^2\): The highest power, we place it first.
  • 7x: The second highest, this linear term comes next.
  • 10: Last is the constant as it has no variable.
Thus, from \(7x - 5x^2 + 10\), we organize it to -5\(x^2\) + 7x + 10. Rearranging doesn't change the value of the expression; it simply makes it cleaner and easier to work with.
Simplifying Algebraic Expressions
Simplifying an algebraic expression involves combining like terms or reducing the terms wherever possible. In the given quadratic expression -5\(x^2\) + 7x + 10, we inspect whether any simplification is feasible.
Since the terms -5\(x^2\), 7x, and 10 are all distinct, i.e., they belong to different categories (quadratic, linear, and constant), there are no like terms to combine. Therefore, no arithmetic simplification can be performed here.
It's crucial to understand that an expression is in its simplest form when all like terms are combined, and no further reduction is possible. Hence, -5\(x^2\) + 7x + 10 is already as simplified as it can be.

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