Problem 56 - Algebra A Combined Function

Question

Multiply. $$ (8 y+9)(8 y-9) $$

Step-by-Step Solution

Answer

The product is \(64y^2 - 81\).

Step 1: Identify the formula

This exercise is a multiplication of two binomials in the form \((a+b)(a-b)\). It fits the difference of squares pattern \(a^2 - b^2\).

Step 2: Assign values to variables

Identify the values of \(a\) and \(b\) from the expression \((8y+9)(8y-9)\). Here, \(a = 8y\) and \(b = 9\).

Step 3: Apply the difference of squares

Plug the values of \(a\) and \(b\) into the difference of squares formula: \((a^2 - b^2)\). This gives us \((8y)^2 - 9^2\).

Step 4: Calculate \((8y)^2\)

Square \(8y\) to get \((8y)^2 = 64y^2\).

Step 5: Calculate \(9^2\)

Square the number \(9\) to get \(9^2 = 81\).

Step 6: Subtract the squares

Subtract \(9^2\) from \((8y)^2\): \(64y^2 - 81\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a valuable algebraic tool. It is used to simplify expressions or solve equations. The difference of squares is based on the identity:

• \(a^2 - b^2 = (a+b)(a-b)\)

This formula signifies that the product of the sum and the difference of the same two terms will always result in the difference of their squares.
This formula applies wherever we see binomials in the form

• \((a+b)(a-b)\)

For example, when you have

• \((8y+9)(8y-9)\)

The task is to identify and apply the difference of squares pattern. Recognize

• \(a = 8y\)
• \(b = 9\)

Then, simply plug these into the formula. Multiplying directly often involves complex calculations. However, recognizing the pattern simplifies work into straightforward squares and subtraction. The difference of squares provides both efficient calculation and deep understanding of algebraic structures.

Binomials

Binomials are expressions that contain two terms, often separated by a plus "+" or minus "-" sign. A binomial might look like

• \((a+b)\)
• \((a-b)\)

In our exercise, both

• \( (8y+9)\)
• \( (8y-9)\)

are binomials. Binomials are foundational in algebra because they serve as building blocks for more complex polynomial expressions.
The beauty of binomials comes from their interactions. This includes their role in identities such as the difference of squares, and being part of bigger polynomial structures. Whether adding, subtracting, or multiplying, understanding how binomials behave is crucial. When handling binomials, always identify their individual terms. Determine how they interact, especially when linked by operations like multiplication. This helps break down complicated problems into manageable pieces.

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that contain more than one term. Binomial multiplication is a basic form of this process. Key techniques include the distributive property and special identities.
The traditional method of polynomial multiplication is to apply the distributive property to each term of the polynomial. For the expression

• \((8y+9)(8y-9)\)

recognizing the difference of squares simplifies things.
Instead of performing term-by-term multiplication, apply the identity:

• \((a+b)(a-b) = a^2 - b^2\)

By identifying this special form, we avoid complicated arithmetic. We substitute and simply compute squares and subtraction.
This approach showcases the power of recognizing patterns in algebra. It is not just execution, but understanding that leads to accuracy and efficiency. Polynomial multiplication becomes less intimidating, more about recognizing these hidden relationships than computing intricate operations directly.