Problem 18 - Elementary Algebra

Question

Multiply. (Assume all denominators are nonzero.) $$ 12 y 2 y 3(2 y-1) \cdot(2 y-1) 3 y $$

Step-by-Step Solution

Answer

The product is \( 864 y^5 - 864 y^4 + 216 y^3 \).

Step 1: Expand the Product

First, write the expression as a multiplication of all factors, which can be combined:\[ 12 \cdot y \cdot 2 \cdot y \cdot 3 \cdot (2y - 1) \cdot (2y - 1) \cdot 3 \cdot y \]

Step 2: Simplify by Multiplying Constants

Multiply all constants together: \[ 12 \cdot 2 \cdot 3 \cdot 3 = 216 \] This simplifies the expression to:\[ 216 \cdot y \cdot y \cdot (2y - 1) \cdot (2y - 1) \cdot y \]

Step 3: Combine Powers of y

Multiply all instances of \( y \):\[ y \cdot y \cdot y = y^3 \] Now the expression is:\[ 216 \cdot y^3 \cdot (2y - 1) \cdot (2y - 1) \]

Step 4: Square the Binomial

Compute \( (2y - 1)^2 \) using the formula \((a - b)^2 = a^2 - 2ab + b^2\):\[ (2y - 1)^2 = (2y)^2 - 2 \cdot 2y \cdot 1 + 1^2 = 4y^2 - 4y + 1 \]

Step 5: Final Multiplication

Multiply the constant \(216 y^3\) by each term of the expanded binomial:\[ 216 y^3 (4y^2 - 4y + 1) = 216 \times 4 y^5 - 216 \times 4 y^4 + 216 y^3 \] which simplifies to:\[ 864 y^5 - 864 y^4 + 216 y^3 \]

Key Concepts

Multiplying Polynomials

Multiplying polynomials involves distributing each term in the first polynomial by every term in the second polynomial. In our example, we start with simplifying the polynomial expression:

• First, note the numeric coefficients like 12, 2, and 3. You multiply these numbers directly to simplify calculations later on.
• Each term goes through this process, and it's important to simplify step by step.

In general, you are using the distributive property of algebra for these calculations. A helpful approach is to start with the constants and then handle each variable part separately.
For many students, lining up terms to ensure none are missed can also help maintain accuracy.

Binomial Expansion

The expansion of a binomial, particularly when squared, follows a pattern that is derived from the formula \[(a - b)^2 = a^2 - 2ab + b^2\].
This method allows for the systematic expansion of the expression without missing any components.

• In the example, the expression \((2y - 1)^2\) is expanded according to this pattern.
• Each part of the binomial contributes to different terms in the final result.
• When you expand, make sure to calculate \((a)^2\), \(-2ab\), and \((b)^2\) accurately.

As you work through the steps, keeping each term clearly separated and simplified can ensure that you don't overlook any coefficients or signs.

Powers of Variables

Understanding how to handle powers of variables is crucial, especially when dealing with multiple instances of a variable.
In our exercise, you encounter powers of \(y\) like \(y\cdot y = y^2\) and \(y^3\).

• The rules of exponents tell us that when you multiply powers of the same base, you add the exponents.
• Thus, \(y\cdot y\cdot y = y^{1+1+1} = y^3\).

Accurate multiplication of variables involves accounting for each exponent separately to avoid mistakes.
Remember, keeping track of these powers during multiplication helps ensure all terms are correctly simplified.