Problem 62 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ -z(z-2)+3 z^{2}+5 $$

Step-by-Step Solution

Answer

The simplified expression is \( 2z^{2} + 2z + 5 \)

Step 1: Distribution of Negative Sign

Distribute the negative sign in \( -z(z-2) \) to both \(z\) and \(-2\). This gives: \(-z^2 + 2z\).

Step 2: Write All Terms

Now write down all the terms by also including '3z^2' and '5' from the original expression. The expression becomes: \(-z^2 + 2z + 3z^2 + 5\).

Step 3: Combine Like Terms

Combine like terms, this involves adding up the coefficients of like terms. The coefficient of \(z^{2}\) is \(-1 + 3\) and the only other term is '5'. Thus the expression simplifies to \(2z^{2} + 2z + 5\)

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of transforming a mathematical expression into its simplest form while maintaining equality. This is a vital skill in algebra, as it helps streamline complex problems into more manageable parts.
When simplifying expressions, it's important to remember to eliminate unnecessary parentheses and combine like terms. By doing so, the expression is reduced to its basic components, making it easier to understand and solve.
Let's start with removing any symbols of grouping, such as parentheses. For example, consider the expression \(-z(z-2) + 3z^2 + 5\). The parentheses indicate a need to multiply or distribute, which is our next focus.

Combining Like Terms

Combining like terms is an essential skill in the realm of algebra. It involves merging terms in an expression that have identical variable parts (e.g., terms like \(z^2\) in an equation are considered 'like' terms).
A critical aspect of combining like terms is to correctly identify these terms within a mathematical expression. For instance, in the expression \(-z^2 + 2z + 3z^2 + 5\), the terms \(-z^2\) and \(3z^2\) are like terms because they share the variable part \(z^2\).
After identifying like terms, the coefficients—numbers in front of the variable parts—are added or subtracted accordingly. So, solving for the expression:\(-1z^2 + 3z^2\) results in \(2z^2\). Remember, only like terms can be combined to provide a simplified expression.

Distribution in Algebra

Distribution in algebra refers to spreading or distributing one term across terms inside a grouping symbol like parentheses. This action is rooted in the distributive property of multiplication over addition, a fundamental property in mathematics.
Take the expression \(-z(z-2)\) as an example. To distribute, multiply each term inside the parentheses by \(-z\). You end up with two separate terms: \(-z \times z\), which equals \(-z^2\), and \(-z \times -2\), which equals \(2z\).
Use distribution whenever an expression has parentheses followed by a multiplicative factor (like \(-z\) here). Properly distributing and removing these grouping symbols allows us to simplify the expression in successive steps, ultimately achieving a simpler form for effective solving.