Problem 20 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (y+4) 5 $$

Step-by-Step Solution

Answer

The expression \( (y+4) 5 \) without parentheses is \( 5y + 20 \)

Step 1: Identify terms inside and outside parentheses

The given expression is \( (y+4) 5 \). Here, 'y' and '4' are inside the parentheses and '5' is outside the parentheses.

Step 2: Apply the Distributive Property

Multiply each term inside the parentheses with the term outside the parentheses. This is how the distributive property works.\nStart with \( y \times 5 + 4 \times 5 \)

Step 3: Perform the multiplication

Now multiply the terms: \( 5y + 20 \)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. Variables can stand for numbers we do not yet know, and they let us create general rules for manipulating these numbers. For instance, in the expression \( (y+4) 5 \), \( y \) represents an unknown variable that can take on different values.

In algebra, we often work with such expressions to perform various operations, such as addition, subtraction, multiplication, and division. The operation we choose depends on what we need to find out or simplify. In the exercise, we are multiplying the entire algebraic expression \( (y+4) \) by \( 5 \) to eliminate the parentheses and simplify the expression.

Multiplying Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, arranged in terms of powers of a variable or variables. Multiplication of polynomials involves using the distributive property to multiply each term of one polynomial by each term of the other polynomial.

In our example, although \( y+4 \) is not a polynomial in the standard form, we treat it as such by applying the same principles to multiply by \( 5 \) outside the parentheses. We distribute the \( 5 \) across the two terms inside the parentheses, \( y \) and \( 4 \), resulting in two separate products: \( 5y \) and \( 5 \times 4 \).

Multiplication Step By Step

• Identify each term in the polynomial
• Use the distributive property to multiply each term
• Write the product for each multiplication

Completing these steps results in a simplified expression without parentheses.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as simple as possible. This means removing parentheses, combining like terms, and reducing fractions when applicable. Simplification helps make the expression clearer and easier to work with.

Using the distributive property allows us to remove parentheses as we did in the original exercise by multiplying \( (y+4) 5 \). After distributing the multiplication and combining the terms, we get \( 5y + 20 \), which is a simpler form of the original expression. Simplification doesn't change the value of the expression; it just makes it more straightforward to evaluate or compare.

Why Simplify?

• It reduces the complexity of calculations
• It reveals the structure of the expression
• It makes it easier to plug in values for variables

Remember, always check for any opportunity to combine like terms after distributing and multiplying to ensure the expression is fully simplified.