Problem 43 - Algebra A Combined Function

Question

Multiply. $$ (3 x-1)(3 x+1) $$

Step-by-Step Solution

Answer

The product is \(9x^2 - 1\).

Step 1: Recognize the Type of Expression

The expression \((3x-1)(3x+1)\) is a product of conjugates. Conjugates are expressions of the form \((a-b)(a+b)\).

Step 2: Use the Formula for Conjugates

The product of conjugates is given by the formula \((a-b)(a+b) = a^2 - b^2\). In this problem, \(a = 3x\) and \(b = 1\).

Step 3: Substitute into the Formula

Substitute the values of \(a\) and \(b\) into the formula: \[(3x-1)(3x+1) = (3x)^2 - (1)^2.\]

Step 4: Compute Each Term

Calculate \((3x)^2\) and \((1)^2\): - \((3x)^2 = 9x^2\).- \((1)^2 = 1\).

Step 5: Simplify the Expression

Combine the terms to simplify: \[9x^2 - 1.\]

Key Concepts

Product of Conjugates

When you come across an expression like \[(3x-1)(3x+1),\] you are dealing with what is called a "product of conjugates." Conjugates are special pairs of binomials, expressed in the pattern \[(a-b)(a+b).\] This means that the expressions have identical terms with opposite signs between them.
The beauty of working with conjugates lies in their simplicity, because multiplying them leads to a straightforward result. Instead of expanding each term and combining, you can use an important algebraic formula:

• \((a-b)(a+b) = a^2 - b^2\).

In our example, \(a = 3x\) and \(b = 1\). By applying this formula, you can skip the longer route of multiplying binomials and directly derive the simplified result.

Multiplying Binomials

Though the formula for conjugates simplifies our work, understanding how to multiply binomials in general is a vital skill. Binomials are expressions with two terms, and multiplying them requires you to use the distributive property. This is often remembered with the acronym FOIL:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

For instance, let's consider multiplying \[(3x-1)(3x+1)\] as you would two generic binomials:

• First: \(3x \times 3x = 9x^2\)
• Outer: \(3x \times 1 = 3x\)
• Inner: \(-1 \times 3x = -3x\)
• Last: \(-1 \times 1 = -1\)

Notice how the middle terms, \(3x\) and \(-3x\), cancel each other out, leaving only \(9x^2 - 1\) in the end. This matches our previous, simpler approach using the product of conjugates.

Simplifying Expressions

Simplifying expressions is about combining like terms to make an expression as concise as possible. After using the product of conjugates formula, we landed with \[9x^2 - 1.\]This step might seem trivial, but it is crucial; it ensures that you present your final answer neatly and logically.
Here are some tips for simplification:

• Always look for terms that can be combined, such as similar variables or constants.
• Check your work to ensure no like terms are left uncombined.
• Ensure your coefficients are as reduced as possible, avoiding any common factors.

The goal is to not just solve but also to express your solution in the simplest form possible, which makes it easier to understand and work with in further calculations.