Problem 48 - Algebra A Combined Function

Question

Multiply. $$ \left(3 x^{2}+1\right)\left(3 x^{2}-1\right) $$

Step-by-Step Solution

Answer

\(9x^4 - 1\)

Step 1: Understand the Problem

We are asked to multiply two binomials: \((3x^2 + 1)\) and \((3x^2 - 1)\). These are in the form of \((a+b)(a-b)\), which is recognizable as a difference of squares.

Step 2: Remember the Formula

The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\). In this expression, \(a = 3x^2\) and \(b = 1\).

Step 3: Apply the Formula

Substitute \(a\) and \(b\) into the difference of squares formula: \((3x^2)^2 - 1^2\).

Step 4: Calculate Each Square

Calculate \((3x^2)^2\) which is \(9x^4\) and \(1^2\) which is \(1\).

Step 5: Subtract the Squares

Subtract the two squares: \(9x^4 - 1\). This is the result of the multiplication.

Key Concepts

Multiplying Binomials

Multiplying binomials can be simplified by recognizing patterns in algebraic expressions. When given binomials in the form \((a + b)(a - b)\), you can apply the **difference of squares** formula. This formula states that multiplying such binomials results in \(a^2 - b^2\). Basically, multiplication here boils down to squaring each term and then subtracting.

Let's break it down: if you have two binomials that are identical except for the sign between their terms, like \((3x^2 + 1)(3x^2 - 1)\), directly use \((3x^2)^2 - 1^2\) to find the result quickly.

• First, square each individual part:
  • \((3x^2)^2\) becomes \(9x^4\)
  • \(1^2\) becomes \(1\)
• Subtract the square of the second term from the first: \(9x^4 - 1\)

This example illustrates that you can streamline the process of multiplying binomials by leveraging known algebraic identities. Instead of individually multiplying each term and simplifying, spotting the pattern makes it easier.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations that form a concise way to represent mathematical relationships. They are the building blocks in algebra and key to solving real-world problems. The expression \((3x^2 + 1)(3x^2 - 1)\) is made of two binomials, showcasing how variables and constants come together in applied math.

The expression simplifies our multiplication task by using a formula for a specific type of polynomial multiplication, thus avoiding lengthy calculations. Algebraic expressions like these leverage key arithmetic operations:

• **Addition and Subtraction:** Combining like or unlike terms.
• **Multiplication:** Often involving distributive property or specialized formulas like the difference of squares.
• **Exponentiation:** A fundamental operation in expressions which involves powers, like \((3x^2)^2\).

This exercise confirms how algebraic expressions enable efficient problem-solving strategies by using established formulas.

Polynomials

Polynomials are expressions containing variables raised to whole number exponents. They are fundamental in algebra due to their power in describing curves and patterns. In the problem \((3x^2 + 1)(3x^2 - 1)\), each binomial is part of the polynomial family. Understanding their properties helps simplify complex mathematical operations.

Each part of a polynomial is termed a 'term,' which consists of a coefficient (a constant) and a variable part. In our case, \(3x^2\) is a single term of the binomial labeled by the coefficient 3 and variable part \(x^2\). This specific exercise narrows in on a special case of polynomial multiplication called the "difference of squares."

• **Terms:** Unit segments like \(3x^2\) or \(1\) are the building blocks of a polynomial.
• **Degree:** The highest power exponent in a polynomial, here it initially appears in \(x^2\) before manipulation.
• **Operations:** Applying addition, subtraction, multiplication, or special patterns.

Through understanding polynomials and their interactions, we better appreciate how they model change and relationships in quantitative situations.