Problem 46 - Algebra A Combined Function

Question

Multiply. $$ \left(a^{2}+6\right)\left(a^{2}-6\right) $$

Step-by-Step Solution

Answer

The product is \(a^4 - 36\).

Step 1: Identify the Pattern

Recognize that the expression \((a^2 + 6)(a^2 - 6)\) fits the pattern of a difference of squares, which is \((x+y)(x-y) = x^2 - y^2\). In this expression, let \(x = a^2\) and \(y = 6\).

Step 2: Apply Difference of Squares Formula

Use the difference of squares formula: \((x + y)(x - y) = x^2 - y^2\). Substitute \(x = a^2\) and \(y = 6\) into the formula.

Step 3: Perform the Calculation

Calculate \(x^2 - y^2\), which is \((a^2)^2 - 6^2\). This becomes \(a^4 - 36\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication is a fundamental concept in algebra. It is the process of multiplying two or more polynomials together to get a single polynomial.
In this exercise, we are looking at multiplying \( (a^{2}+6) \) and \( (a^{2}-6) \). It is essential to note that each polynomial is composed of terms. In the expression given, the terms include \( a^2 \) and constants like \( 6 \).
When multiplying polynomials, each term in one polynomial must be multiplied by each term in the other polynomial. Here are a few steps to simplify the process:

• Identify the terms in each polynomial.
• Apply the distributive property to multiply each term.
• Simplify and combine like terms, if needed.

In this specific problem, the multiplication process is simplified by recognizing that it follows a difference of squares pattern, which we'll explore next.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the basis of algebra and are used to represent real-world situations in mathematical form.
The expression given in the exercise, \( (a^2 + 6)(a^2 - 6) \), is an algebraic expression because it consists of variables \( a \), constants like \( 6 \), and the operations of addition, subtraction, and multiplication.
Understanding how to properly manipulate these expressions is crucial for problem-solving in algebra. Here are some tips for working with algebraic expressions:

• Know the components: terms, coefficients, and powers.
• Apply operations according to algebraic rules.
• Look for patterns, like factoring, to simplify expressions.

Recognizing and working with these components allows for simplified solutions and better understanding when facing complex equations.

Factoring Techniques

Factoring techniques involve writing a polynomial as a product of its factors. This process is helpful in solving equations, simplifying expressions, and making polynomial multiplication more manageable.
One common factoring technique is the "difference of squares" formula. This formula states that the product of two conjugates \( (x + y)(x - y) \) can be factored into \( x^2 - y^2 \).
In our exercise, the expression \( (a^2 + 6)(a^2 - 6) \) fits perfectly into this difference of squares pattern. By identifying \( x = a^2 \) and \( y = 6 \), we can express it as \( x^2 - y^2 \) or \( (a^2)^2 - 6^2 \), leading to the simplified expression \( a^4 - 36 \).
Using factoring techniques like this quickly simplifies multiplication and helps with efficient problem-solving in algebra.