Problem 45 - Algebra A Combined Function

Question

Multiply. $$ \left(x^{2}+5\right)\left(x^{2}-5\right) $$

Step-by-Step Solution

Answer

The product of \((x^{2}+5)(x^{2}-5)\) is \(x^4 - 25\).

Step 1: Identify the Expression Format

The expression to be multiplied is \((x^2 + 5)(x^2 - 5)\). This expression is in the format of \((a + b)(a - b)\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a + b)(a - b) = a^2 - b^2\). Apply this to our expression, using \(a = x^2\) and \(b = 5\).

Step 3: Compute the Squares

Calculate \(a^2 = (x^2)^2 = x^4\) and \(b^2 = 5^2 = 25\).

Step 4: Substitute and Simplify

Substitute the values back into the formula: \((x^2)^2 - 5^2 = x^4 - 25\). This is the simplified result of multiplying the two expressions.

Key Concepts

Algebra

Algebra is all about using letters to represent numbers in equations and expressions. It's like a system or a language that helps us solve mathematical problems. One of the foundational ideas in algebra is working with variables, like \(x\) in our exercise. Variables can represent any number, which is what makes algebra incredibly versatile.

Understanding algebra involves:

• Learning how to manipulate expressions by following rules and procedures.
• Using techniques like factoring, expanding, and simplifying.
• Recognizing patterns and applying formulas.

For example, in the expression \((x^2 + 5)(x^2 - 5)\), we use the principles of algebra to apply the _difference of squares_ formula, a handy pattern that simplifies our work. This idea becomes powerful as you learn to solve equations and work through more complex mathematics.

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that contain variables raised to whole number powers. The expressions are written as sums or differences of these powers. In our exercise, \((x^2 + 5)(x^2 - 5)\), we're dealing with polynomials.

When multiplying polynomials, we apply basic multiplication, but with each term. However, in this case, our expression fits a unique pattern known as the difference of squares. This pattern allows us to use the formula \((a + b)(a - b) = a^2 - b^2\) to simplify the task without doing step-by-step multiplication. Typically, you'd multiply out each term in each polynomial, but recognizing patterns can make the process much quicker.

• Identify each part: Here, \(a\) is \(x^2\) and \(b\) is 5.
• Understand the formula: The difference of squares lets you multiply swiftly as it provides a ready-made solution.

This simplification is what makes understanding polynomial multiplication so effective.

Expression Simplification

Expression simplification is all about making an expression easier to understand or work with. We remove unnecessary complexity, find patterns, and reduce expressions to simpler forms.

For the expression \((x^2 + 5)(x^2 - 5)\), the process involves:

• Seeing the pattern: Recognize it as a difference of squares.
• Calculating the individual squares: \((x^2)^2\) for \(a\) and 5 for \(b\).
• Putting it together: Substituting back, \(x^4 - 25\) becomes the simplified form.

The goal is to make math manageable and efficient. When you apply strategies like factorization or detection of special patterns, the task becomes easier and often much faster. Simplifying expressions is a key skill as it streamlines problem-solving in algebra and beyond.