Problem 36 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -s(7+s) $$

Step-by-Step Solution

Answer

The expression \(-s(7+s)\) simplifies to \(-7s - s^2\) using the distributive property.

Step 1: Expand the Parentheses

Start by multiplying every term inside the parentheses \((7+s)\) by the term that's outside the parentheses \(-s\). This means distributing \(-s\) across the terms within the parentheses.

Step 2: Multiply -s and 7

Multiply -s by 7. This will give you \(-7s\).

Step 3: Multiply -s and s

Next, multiply -s by s. Remember that -s times s equals \(-s^2\).

Step 4: Write the final expression

Write the two products you calculated step by step in one expression. The final expression is: \(-7s - s^2\).

Key Concepts

Expanding Parentheses

Expanding parentheses is a foundational technique in algebra for simplifying expressions. The distributive property allows us to remove parentheses by distributing a factor outside the parentheses across each term inside. For instance, with the expression \( -s(7+s) \), the factor outside the parentheses is \( -s \).

To expand the parentheses, we apply \( -s \) to each term inside the parentheses individually. This is like sharing a piece of information with every person in a group. Just as everyone in the group gets the news, every term in the parentheses gets multiplied by \( -s \). Hence, \( -s \) multiplies with \( 7 \) to give \( -7s \) and \( -s \) multiplies with \( s \) to give \( -s^2 \). This process changes the original expression \( -s(7+s) \) into \( -7s - s^2 \) by expanding out the parentheses.

Multiplying Polynomials

Multiplying polynomials involves applying the distributive property multiple times. Polynomials are like several numbers huddled together, and when we multiply them, we are essentially swirling them into a dance of multiplication. Take the expression from the previous concept, where we only had one term outside and two inside the parentheses. A polynomial multiplied by a monomial is a simpler case of the general principle.

When dealing with polynomials consisting of several terms, we still multiply each term of one polynomial by every term of the other polynomial. It's like a handshake at a networking event—every term 'meets' every other term. Just remember to combine like terms at the end, which means gathering all the terms with the same exponent and combining them to simplify the expression further.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making them as neat and as manageable as possible. Following the expansion and multiplication, simplifying is like tidying up the room after a grand party; it’s essential to maintain clarity and understanding. For the given expression \( -7s - s^2 \), simplification doesn't further condense the expression since there are no like terms to combine.

However, in more complex cases, simplification involves combining like terms (terms with the same variable and exponent) and reducing fractions or expressions to their lowest terms. It’s important to pay attention to the signs of the terms and properly apply the basic arithmetic operations—addition, subtraction, multiplication, and division—as required to tidy up the expression. Keep in mind that the ultimate goal is to have the simplest form that is equivalent to the original expression.