Question

Apply the distributive property. $$2 y(8-7 y)$$

Short answer

The result of applying the distributive property to \(2y(8-7y)\) is \(16y - 14y^2\).

Step-by-step solution

Distribute \(2y\) across \(8 - 7y\)

Multiply \(2y\) by each term inside the parentheses: \(2y*8\) and \(2y*(-7y)\).

Simplify Multiplications

Carry out the multiplication operations: \(2y*8 = 16y\) and \(2y*(-7y) = -14y^2\).

Key concepts

Understanding Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and represent a combination of numbers, variables, and mathematical operators such as addition, subtraction, multiplication, and division. For example, the expression 2y(8 - 7y) consists of numbers (2, 8, -7), a variable (y), and an implied multiplication between 2y and the parenthetical expression.

An essential skill when working with algebraic expressions is recognizing structures that can be simplified using algebraic properties. Among these is the distributive property, which allows us to multiply a single term by each term inside a parenthesis and then add the results together. This property connects multiplication with addition, showing that multiplication can be 'distributed' over addition within an expression. Let's have a closer look at how this unfolds in practice.

Multiplication of Polynomials

When multiplying polynomials, such as in the exercise 2y(8 - 7y), you are dealing with a foundational algebraic process: multiplying each term of one polynomial by every term of the other. This operation expands an algebraic expression, leading to a simplified form.

Multiplication of polynomials follows the distributive property, as illustrated in the textbook solution. The first element outside the parenthesis, 2y, is multiplied by each of the terms inside, which are 8 and -7y.

Let's dissect it:

• Multiplying 2y by 8 yields 16y.
• Multiplying 2y by -7y gives us -14y^2 due to the 'like terms' being combined (i.e., the y's).

This method is vital for dealing with more complex polynomial expressions, as it provides a systematic approach for expansion and simplification.

Simplifying Expressions

The process of simplifying expressions involves reducing them to their most basic form without changing their value. This usually includes combining like terms, distributing coefficients, and performing arithmetic operations. In the context of the problem at hand, we simplify the expression by multiplying and then combining like terms.

To thoroughly simplify the given expression 2y(8 - 7y), we applied the distributive property and multiplication, and now we arrive at the terms 16y and -14y^2.

Given that these terms are not like terms (one is a linear term, and the other is a quadratic term), we cannot combine them further. Thus, our simplified expression is 16y - 14y^2. In other cases, if the resulting terms were like terms, the simplification process might include another step – combining those terms to reach a final simplified form. Bringing expressions to their simplest form is critical for effectively solving algebraic equations and comparing algebraic quantities.