Problem 73 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ \left(x^{2}+10\right)\left(x^{2}-10\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(x^4 - 100\).

Step 1: Identify the Form

Recognize that the given expression \((x^2+10)(x^2-10)\) is a difference of squares. The structure \((a+b)(a-b)\) is a difference of squares formula, where \(a = x^2\) and \(b = 10\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to simplify the expression. Substitute \(a = x^2\) and \(b = 10\) into the formula, getting \((x^2)^2 - 10^2\).

Step 3: Calculate the Squares

Calculate the squares of the terms: \((x^2)^2\) equals \(x^4\) and \(10^2\) equals 100. Substitute these values back into the expression, giving \(x^4 - 100\).

Step 4: Simplify the Result

The expression \(x^4 - 100\) is already in its simplest form. This is the simplified result of the original problem.

Key Concepts

Polynomial Multiplication

When multiplying polynomials, you are essentially distributing each term from one polynomial to every term in another polynomial. In general, it involves extending the distributive property over each pair of terms in the two polynomials.
For our exercise, however, the expression \[ (x^2+10)(x^2-10) \] is in a special form known as the "difference of squares", so the regular rules of polynomial multiplication can be simplified using a specific formula.
The difference of squares formula, \( (a+b)(a-b) = a^2 - b^2 \), is a handy shortcut when dealing with expressions of this type. Instead of multiplying each term individually, the formula lets you immediately jump to the simplified form.

Algebraic Expressions

Algebraic expressions, like \( x^2+10 \) and \( x^2-10 \), are made up of variables and constants combined with operations such as addition, subtraction, multiplication, and division.
These expressions are foundational in algebra and are used to model real-world situations. They can often be combined or manipulated to find simplified results or to solve equations.
In our exercise, each of the two parts of the expression, \( x^2+10 \) and \( x^2-10 \), demonstrates an important algebraic concept: they represent binomials (expressions with two terms). Recognizing the structure of such expressions helps us to see patterns like the difference of squares.

Factoring Techniques

Factoring is a powerful technique in algebra that involves rewriting an expression as a product of its factors. It often makes complex expressions simpler to work with.
In the given exercise, the expression \((x^2+10)(x^2-10)\) is already expressed in its factored form. By recognizing it as a difference of squares, we used the formula \(a^2 - b^2\) to represent the factors accurately.
The process of recognizing and applying such techniques allows for swift simplification and manipulation of algebraic expressions, especially when preparing them for further operations or solving related equations. Factoring not only aids in simplification but also in revealing hidden relationships between terms.