Problem 40 Simplify. (Assume all denominato... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 7: Problem 40

Simplify. (Assume all denominators are nonzero.) $$ 2-52 x-3 x 24 x+3 $$

Short Answer

Expert verified

The simplified expression is $-28x - 1$.

Step by step solution

Identify Terms

This expression contains both constant and variable terms: $2$, $-52x$, $-3$, and $24x$.

Combine Like Terms

First, identify and combine the constant terms: $2$ and $-3$. Then, find the coefficient of the $x$ terms: $-52x + 24x$.

Simplify Constant Terms

Simplify the constant terms: $2 - 3 = -1$.

Simplify Variable Terms

Simplify the $x$ terms by subtracting their coefficients: $-52x + 24x = -28x$.

Combine All Simplified Terms

Combine the simplified constant term with the variable term to achieve the final expression: $-28x - 1$.

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Key Concepts

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

Combining Like Terms

In algebra, combining like terms is a fundamental skill that simplifies expressions. Think of it like cleaning up clutter by grouping similar items together. When we say "like terms," we refer to terms that have the same variable raised to the same power. For example, in the expression $-52x + 24x$, both terms contain the variable $x$. Hence, they are like terms.
To combine them, we simply add or subtract the coefficients (the numbers in front of the variables). In our example, subtract 52 from 24 to get $-28x$.

Look for terms with the same variable and power.
Add or subtract their coefficients.

This process reduces the complexity of expressions, helping us see the essentials more clearly.

Polynomial Expressions

Polynomial expressions are sums and differences of terms, each consisting of a coefficient multiplied by a power of a variable. They can look like a mix of constants and variables, like our expression: $2 - 52x - 3 + 24x$. Understanding polynomials involves knowing their terms, degrees, and coefficients.
Each term within a polynomial is separated by a plus (+) or minus (-) sign. Polynomials can have many terms or just a few.

"Monomials" are single terms, like $-52x$.
"Binomials" have two terms, like $-52x + 24x$.
"Trinomials" have three terms, and so on.

Simplifying these expressions often involves combining like terms, which streamlines the polynomial and makes it easier to work with.

Variables and Constants

Variables and constants form the building blocks of algebraic expressions. A variable, often represented by symbols like $x$, $y$, or $z$, is an unknown quantity that can change.
On the other hand, a constant is a fixed value. In our exercise, the numbers 2 and -3 are constants, while $-52x$ and $24x$ include variables ($x$) with specific coefficients.
So, what's the difference?

Variables can vary, as their value isn't fixed. They act as placeholders for numbers.
Constants stay the same. They don't change, adding stability to expressions.

Understanding this distinction is crucial for simplifying expressions and solving equations, as it helps determine which parts of the expression can change and which stay put.

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91影视

Simplify. (Assume all denominators are nonzero.) $$ 2-52 x-3 x 24 x+3 $$

Short Answer

Step by step solution

Identify Terms

Combine Like Terms

Simplify Constant Terms

Simplify Variable Terms

Combine All Simplified Terms

Key Concepts

Combining Like Terms

Polynomial Expressions

Variables and Constants

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