Problem 53 - Algebra for College Students

Question

Perform the indicated operations. $$\left(7 x^{2}-x-4\right)-\left(9 x^{2}-10 x+8\right)+\left(12 x^{2}+4 x-6\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(10x^2 + 13x - 18\).

Step 1: Distribute the Signs

To begin, distribute the negative sign across the terms in the second polynomial. This changes the signs of all the terms in that polynomial:\[7x^2 - x - 4 - (9x^2 - 10x + 8) + (12x^2 + 4x - 6) = 7x^2 - x - 4 - 9x^2 + 10x - 8 + 12x^2 + 4x - 6\]

Step 2: Combine Like Terms

Next, group and combine the like terms, which are the terms with the same power of \(x\):1. Combine the \(x^2\) terms: \(7x^2 - 9x^2 + 12x^2 = 10x^2\).2. Combine the \(x\) terms: \(-x + 10x + 4x = 13x\).3. Combine the constant terms: \(-4 - 8 - 6 = -18\).

Step 3: Write the Simplified Expression

Now write down the simplified expression obtained by combining like terms:\[10x^2 + 13x - 18\]

Key Concepts

Combining Like Terms

In algebra, combining like terms is a technique used to simplify mathematical expressions. When we talk about "like terms," we mean terms that have the same variables raised to the same power.
To combine them, add or subtract the coefficients. Coefficients are the numerical parts of a term. This process ultimately reduces the number of terms in the expression, making it simpler and easier to interpret.
For example, consider the expression \(7x^2 - 9x^2 + 12x^2\). All these terms are like terms because they all have the variable \(x^2\). To combine them, you add or subtract the coefficients:

• Start with \(7\)
• Subtract \(9\) (from \(-9x^2\))
• Add \(12\) (from \(12x^2\))

This simplifies to \(10x^2\). The same rule applies to linear terms like \(-x + 10x + 4x\) and constant terms like \(-4 - 8 - 6\). By combining like terms, you make the expression more manageable.

Distributive Property

The distributive property is a fundamental concept in mathematics that allows us to multiply a sum by a number or expression outside a set of parentheses. It essentially "distributes" this multiplication over each term within the parentheses.
In this exercise, we're dealing with a subtraction that involves distributing a negative sign across a group of terms. This is a direct application of the distributive property:

• When you see a negative sign in front of parentheses, it means you need to change the signs of all the terms inside the parentheses.
• For the expression \(- (9x^2 - 10x + 8)\), distributing the negative sign makes it \(-9x^2 + 10x - 8\).

Using this property ensures that each term is properly accounted for, which is crucial for attaining the correct simplified expression.
Remember, distributing doesn't only apply to negatives; it applies to any multiplication involving a sum within parentheses.

Simplifying Expressions

Simplifying an expression means rewriting it in its simplest form. This process involves combining like terms and applying properties, like the distributive property, to eliminate unnecessary parts of the expression.
The goal of simplification is to make the expression as concise as possible without changing its value. In this exercise, we engage in several steps to simplify:

• First, distribute negative signs using the distributive property.
• Second, combine like terms such as \(x^2\) terms, \(x\) terms, and constant terms.
• Finally, rewrite the expression with these combined terms to get a cleaner form.

The outcome is the simplified form of the polynomial, \(10x^2 + 13x - 18\). By simplifying, complex expressions become easier to work with and understand, serving as a foundation for further algebraic operations or problem-solving.