Question

Multiply. $$-5 y(2 y-4)$$

Short answer

Question: Multiply the monomial -5y by the binomial (2y - 4) and simplify the result. Answer: -10y^2 + 20y

Step-by-step solution

Apply the Distributive Property

To multiply the monomial \(-5y\) by the binomial \((2y - 4)\), we need to distribute the monomial across the terms inside the binomial. That is, $$-5y(2y - 4) = (-5y)(2y) + (-5y)(-4)$$

Multiply the terms

We'll now multiply the terms from the distributive property. $$(-5y)(2y) + (-5y)(-4) = (-10y^2) + (20y)$$

Write the final answer

Our final answer in simplified form is: $$-10y^2 + 20y$$

Key concepts

Distributive Property

The Distributive Property is a fundamental concept in math that makes it easier to multiply expressions like a monomial and a binomial. In essence, it allows you to distribute, or "spread out," the monomial across the binomial's terms. In our given exercise, we start with

• \(-5y(2y - 4)\)

Here, \(-5y\) is distributed to each term inside the parentheses:

• \((-5y)(2y) + (-5y)(-4)\)

This method breaks down the multiplication into simpler parts, helping solve more complex expressions easily. This strategy is especially useful in Prealgebra, setting the stage for more advanced algebraic techniques.

Monomial

A monomial is an algebraic expression with a single term. It consists of a coefficient (a number) and one or more variables (like \(y\) in our exercise). In

• \(-5y\)

\(-5\) is the coefficient and \(y\) is the variable. Monomials are simple yet powerful, often serving as building blocks for more complex expressions.
This simplicity makes them invaluable, especially in multiplication problems where they are often paired with a binomial to form expressions that can be distributed and simplified.

Binomial

A binomial is a polynomial with exactly two terms, which can be items like \(2y\) and \(-4\) in the exercise's expression

• \(2y - 4\).

These terms are combined using addition or subtraction. The presence of two terms makes a binomial more complex than a monomial, but they're still manageable when approached systematically.
Binomials frequently appear in mathematics, especially in multiplication and factoring tasks, where they're paired with other expressions.

Multiplication

Multiplication in this context involves taking the monomial and multiplying it by each term of the binomial. Let's break down the exercise:

• First, \((-5y)(2y)\) equals \(-10y^2\), achieved by multiplying both coefficients and variables.
• Second, \((-5y)(-4)\) equals \(20y\), involving similar steps, paying attention to multiplying negative numbers.

This multiplication reshapes the original expression into a new polynomial, combining like terms into a cleaner, more simplified expression. This step is crucial for clarity and understanding in algebra.

Simplification

Simplification is the process of making an algebraic expression as concise and clean as possible. In our exercise, the simplified form is

• \(-10y^2 + 20y\).

This form is achieved by neatly combining the multiplied terms from the distributive step. Simplification is important because it makes expressions easier to read and understand, providing a clear solution to mathematical problems.
It highlights the end goal of many algebraic processes, ensuring you present your work accurately and correctly.