Problem 28 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$-4\left(n^{2}+3\right)-\left(2 n^{2}-7\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-6n^2 - 5\).

Step 1: Distribute the negative sign in the first term

To remove the parentheses and distribute the negative sign in the first part of the expression, we multiply \(-4\) by each term inside the parenthesis \((n^{2}+3)\). This gives: \[-4(n^{2}+3) = -4(n^2) - 4(3) = -4n^2 - 12\].

Step 2: Distribute the negative sign in the second term

Now, distribute the negative sign in the second set of parentheses \((-1)(2n^{2}-7)\). This changes the sign of each term inside:\[-(2n^{2}-7) = -1(2n^2) + 1(7) = -2n^2 + 7\].

Step 3: Combine all distributed terms

Now, write all the distributed terms together from both the expressions: \[-4n^2 - 12 - 2n^2 + 7\].

Step 4: Combine like terms

Combine the like terms by adding or subtracting the coefficients of similar terms (the \(n^2\) terms and the constant terms): - Combine \(-4n^2\) and \(-2n^2\): \(-4n^2 - 2n^2 = -6n^2\)- Combine the constant terms \(-12\) and \(+7\): \(-12 + 7 = -5\).Thus, the expression becomes:\[-6n^2 - 5\].

Key Concepts

Simplifying Expressions

Simplifying expressions involves condensing a longer algebraic equation into its simplest form. This step-by-step process reduces complexity, making it easier to work with.
When simplifying, your main goal is to remove any unnecessary elements, like parentheses, and to combine similar terms. By doing so, you achieve a more concise expression.

• First, address the terms within parentheses.
• Next, apply any arithmetic, like addition or subtraction.
• Finally, gather and combine similar terms.

Through these steps, the expression becomes much more manageable.

Distributive Property

The distributive property is a principle that allows you to multiply a number by a sum or difference within parentheses. In simpler terms, it lets you "distribute" the multiplication across all terms inside the parentheses.
For example, consider the expression:

• \(-4(n^{2}+3)\) is simplified using the distributive property to become two separate terms: \(-4n^2\) and \(-12\).

The power of the distributive property lies in its ability to eliminate parentheses, thereby simplifying the expression.
By reducing the need for parentheses, you can focus on combining and manipulating the terms effectively in the next steps.

Combining Like Terms

Combining like terms involves grouping and simplifying terms in an expression that have the same variable raised to the same power.
To simplify:

• Look for terms with identical algebraic components, such as \(n^2\).
• Once identified, only the numerical coefficients—the numbers in front—are added or subtracted.

In our earlier example,

• \(-4n^2\) was combined with \(-2n^2\) to form \(-6n^2\).
• For constants, the values \(-12\) and \(+7\) were combined to get \(-5\).

This step is crucial as it gives the expression its simplest form, making it easier to understand and solve further.