Problem 34 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ (3-y) y $$

Step-by-Step Solution

Answer

The expression (3-y)y without parentheses and having been distributed properly is \(3y - y^2\).

Step 1: Identify the components of the expression

Identify the two terms within the parentheses, which are 3 and -y, as well as the term outside the parentheses, which is y.

Step 2: Apply the distributive property

Apply the distributive property to the expression. This involves multiplying each term inside the parentheses by the term outside the parentheses. This results in \(3y - y^2\).

Step 3: Write down your final answer

By following the distributive property rule and multiplying each term inside of the parentheses by the y term outside of the parentheses, the expression (3-y)y can be rewritten as \(3y - y^2\) without parentheses.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. For example, in the expression \((3 - y)y\), '3' and '-y' are terms that are combined through subtraction inside the parentheses, and 'y' is the variable outside that is used to multiply the terms within the parentheses.

An important aspect of understanding algebraic expressions like \((3 - y)y\) is recognizing the structure of these terms and how we can apply algebraic properties such as the distributive property to simplify or manipulate them. These expressions can represent various quantities in math problems, and learning to work with them is fundamental in algebra.

Multiplying Polynomials

Multiplying polynomials requires applying the distributive property systematically to every term in the first polynomial by every term in the second polynomial. A polynomial is an algebraic expression that includes constants, variables, and exponents, such as \(y\) in the parentheses of the given expression \((3 - y)y\).

When multiplying, for instance, a monomial (a single term polynomial) by a binomial (a two-term polynomial), you distribute the monomial to each term of the binomial individually. This process often leads to the expansion of the expression and reveals a new polynomial with combined alike terms, if any.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have the exact same variables raised to the same power. For instance, \(3y\) and \(-2y\) are like terms because they both have the same variable 'y' to the first power, whereas \(3y\) and \(-y^2\) are not like terms because their variables are raised to different powers.

After applying the distributive property, like in the expression \(3y - y^2\), you would combine like terms if present to simplify further. In our example, there are no like terms to combine after distributing, so the expression remains as \(3y - y^2\). Understanding how to find and combine like terms is crucial for simplifying expressions and solving algebraic equations.