Problem 37 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 4. $$ 12 y\left(\frac{y+8}{6 y}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(2y + 16\).

Step 1: Distribute

Multiply each term inside the parentheses by the factor outside, which is \(12y\), so we have: \(12y \times \frac{y}{6y} + 12y \times \frac{8}{6y}\).

Step 2: Simplify Each Term

For the first term, \(12y \times \frac{y}{6y}\), divide the \(y\) in the numerator by the \(y\) in the denominator, and reduce \(12/6\) to get \(2y\).For the second term, \(12y \times \frac{8}{6y}\), cancel out the \(y\) in the numerator and denominator, giving you \(\frac{12 \times 8}{6} = 16\).

Step 3: Combine the Terms

Add the simplified terms together: \(2y + 16\). This is the simplified expression after performing the multiplication and simplification.

Key Concepts

Multiplication of Polynomials

When dealing with multiplication of polynomials, like the expression \(12y\left(\frac{y+8}{6y}\right)\), you are essentially using basic arithmetic operations along with variable expressions. Polynomials simplify into expressions that involve sums and products of variables and constants.

• First, identify each component of the polynomial, in this case, \(12y\) outside the parentheses and \(\frac{y + 8}{6y}\) inside.
• Multiply each term in the parenthesis by the term outside — distribute \(12y\) to both \(\frac{y}{6y}\) and \(\frac{8}{6y}\). This is part of the process known as distributing over the sum.

This operation forms the basis of polynomial multiplication and prepares you for the next step, utilizing the distributive property.

Distributive Property

The distributive property is a fundamental algebraic concept that allows you to simplify expressions where a term is multiplied by a sum inside parentheses. For the expression \(12y\left(\frac{y+8}{6y}\right)\), the distributive property helps in moving \(12y\) across each term within the fraction.

• Start by multiplying \(12y\times \frac{y}{6y}\). Break this operation apart: multiply the numerators and the denominators separately. This yields \(\frac{12yy}{6y}\), which simplifies down to \(2y\) once the \(y\)s cancel out and numbers are simplified.
• Next, handle \(12y\times \frac{8}{6y}\). Here, not only do the \(y\)'s cancel, but you also perform division with the constants, leading to \(\frac{12 \times 8}{6} = 16\).

Understanding the distributive property in this context allows you to tackle larger and more complex algebraic expressions with ease.

Simplification Steps

Simplification is the process of reducing an expression to its most concise form, and it's crucial for solving algebraic equations efficiently. After applying polynomial multiplication and the distributive property, you should always aim to simplify the resulting expression.

• A thorough simplification involves canceling like terms, which is evident in \(12y \times \frac{y}{6y}\). Here, \(y\) in both the numerator and the denominator can be canceled, thus reducing complexity.
• Perform arithmetic operations where possible, as seen in \(12y \times \frac{8}{6y}\), which simplifies following the cancelation of \(y\). Here, you simplify the fraction \(\frac{96}{6}\) to get \(16\).
• Finally, once all terms are simplified, combine them as seen in \(2y + 16\). This consolidation forms your simplified algebraic expression.

Each step in simplification reduces the expression further, making it simpler and more straightforward to evaluate or work with in equations.