Problem 11 - College Algebra

Question

For the following exercises, find the sum or difference. $$ \left(12 x^{2}+3 x\right)-\left(8 x^{2}-19\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 + 3x + 19\).

Step 1: Distribute the negative sign

The expression provided is \((12x^2 + 3x) - (8x^2 - 19)\). Start by distributing the negative sign through the second parenthesis. This changes the signs of each term inside the parenthesis: the expression becomes \(12x^2 + 3x - 8x^2 + 19\).

Step 2: Combine like terms

Now, identify and combine like terms in the expression \(12x^2 + 3x - 8x^2 + 19\). For the \(x^2\) terms, combine \(12x^2\) and \(-8x^2\) to get \(4x^2\). The \(x\) term is \(3x\), and the constant term is \(+19\).

Step 3: Write the simplified expression

After combining like terms, write out the simplified expression in standard form: \(4x^2 + 3x + 19\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept in algebra that helps to simplify expressions and make them easier to work with. Like terms refer to terms that have the same variable raised to the same power. To combine them, you simply add or subtract their coefficients.For example, in the expression \(12x^2 + 3x - 8x^2 + 19\), notice how there are two terms involving \(x^2\): \(12x^2\) and \(-8x^2\). These are like terms because both are multiplied by \(x^2\). To combine them, you subtract 8 from 12, giving you \(4x^2\).

• Identify like terms by looking for the same variable with the same exponent.
• Add or subtract the coefficients of these terms.
• Write the result with the common variable and power.

Once combined, the expression becomes much simpler. Here, the simplified version was \(4x^2 + 3x + 19\), making further calculations and interpretations easier.

Simplifying Expressions

Simplifying expressions is all about making them more manageable and easier to understand by performing operations like combining like terms or using other algebraic rules. The goal is to rewrite the expression in its simplest form without changing its value.In practice, simplifying an expression may involve:

• Removing any parentheses by distributing or combining terms like in \(12x^2 + 3x - 8x^2 + 19\).
• Combining all like terms together to reduce the number of terms.
• Arranging terms in descending power order, which is often standard form for polynomial expressions.

Let's say we have an expression like \(12x^2 + 3x - 8x^2 + 19\), simplifying involves finding like terms first and then combining them, leading us eventually to \(4x^2 + 3x + 19\). Remember to check each step as small slips can lead to big errors, especially when dealing with signs and coefficients.

Distributive Property

The distributive property is a useful rule in algebra for simplifying expressions that involve parentheses. This property allows you to multiply a single term by each term inside the parentheses, or in the context of subtraction, it helps in distributing negative signs to all terms within parentheses.For example, consider the expression \((12x^2 + 3x) - (8x^2 - 19)\). Here, the minus sign in front of the second set of parentheses can be considered as multiplying by -1. This changes the terms inside the parentheses:

• The \(8x^2\) becomes \(-8x^2\).
• Likewise, the \(-19\) becomes \(+19\).

Applying the distributive property in expressions like this ensures that each term is affected correctly, maintaining the expression's integrity. After using this property, you should proceed with combining like terms to further simplify the expression resulting in a simplified form: \(4x^2 + 3x + 19\). Be vigilant in applying signs accurately, as mistakes often happen when distributing negative signs.