Problem 43 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ (3 x+4)-(6 x-1) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x + 5\).

Step 1: Distribute the negative sign

To remove the parentheses in the expression \((3x + 4) - (6x - 1)\), we need to distribute the negative sign through the second parentheses. This changes the signs of the terms inside: \(6x\) becomes \(-6x\) and \(-1\) becomes \(+1\). The expression becomes \(3x + 4 - 6x + 1\).

Step 2: Combine like terms

Now, combine the like terms. For the \(x\) terms, combine \(3x\) and \(-6x\) to get \(-3x\). For the constant terms, combine \(4\) and \(1\) to get \(5\). The expression becomes \(-3x + 5\).

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that helps simplify expressions and solve equations. It allows us to distribute one term over others inside parentheses. In simple terms, it involves multiplying the term outside the parentheses by each term inside the parentheses. This property is often written as:

• \(a(b + c) = ab + ac\)

When solving an expression like \((3x + 4) - (6x - 1)\), the distributive property helps us manage the subtraction of a binomial. We treat the subtraction as adding the negative of the second polynomial. This turns \(-(6x - 1)\) into \(-6x + 1\).
Applying this property simplifies the expressions and prepares them for the next steps, such as combining like terms, which we'll discuss next.

Combining Like Terms

Combining like terms is a crucial step to further simplify algebraic expressions. Like terms are those that have the same variable raised to the same power. For example, in the expression from our exercise: \(3x + 4 - 6x + 1\), \(3x\) and \(-6x\) are like terms because both contain the variable \(x\).
To combine them, simply add or subtract their coefficients. In this example, \(3x - 6x\) gives \(-3x\). For constant terms, you add them directly: \(4 + 1 = 5\).
Combining these terms efficiently reduces complexity, leaving us with a simplified expression: \(-3x + 5\). This process makes the mathematical expression much easier to handle, especially when solving equations or further simplifying polynomials.

Polynomials

Polynomials are expressions composed of variables and coefficients, along with operations of addition, subtraction, and multiplication. They are fundamental components of algebra. Terms of a polynomial can vary based on the degree, which is determined by the exponent of the variable.
In our exercise, the expression was a polynomial with terms involving \(x\): \(3x\) and \(-6x\), plus constant terms \(4\) and \(1\). Simplifying polynomials often involves using properties like distribution and combining like terms, helping to clarify and solve more complex algebraic equations.
Understanding polynomials doesn't just help in simplifying expressions but also leads to solving algebraic equations, graphing quadratic functions, and more. Mastery of polynomials paves the way to advanced mathematical concepts and applications.