Problem 113 - College Algebra Essentials

Question

Perform the indicated operations. $$\left(x^{n}+2\right)\left(x^{n}-2\right)-\left(x^{n}-3\right)^{2}$$

Step-by-Step Solution

Answer

The simplified result of the operation is \(6x^n - 13\).

Step 1: Perform the operation within the first brackets

Use the difference of squares formula, \(a^2 - b^2 = (a+b)(a-b)\) with \(a= x^n\) and \(b=2\). Applying this formula to the first part of the expression \(\left(x^{n}+2\right)\left(x^{n}-2\right)\) yields \(x^{2n} - 2^2\), which simplifies to \(x^{2n} - 4\).

Step 2: Perform the operation within parentheses

Applying the formula \((a - b)^2 = a^2 - 2ab + b^2\), for \(x^n\) as \(a\) and \(3\) as \(b\). This gives us \((x^n)^2 - 2 \cdot x^n \cdot 3 + 3^2\), which simplifies to \(x^{2n} - 6x^n + 9\) for the second part of the expression.

Step 3: Perform the subtraction of the two results

Subtract the result from the second part of the expression \(x^{2n} - 6x^n + 9\) from the result of the first part of the expression \(x^{2n} - 4\). This results in \((x^{2n} - 4) - (x^{2n} - 6x^n + 9)\). This simplifies to \(6x^n - 13\) after combining like terms.

Key Concepts

Difference of Squares

Have you ever looked at an algebraic expression and noticed it's like looking at a mathematical sandwich? One piece of lunchmeat here, one slice of cheese there, and poof – you have your meal ready to go! That's sort of like the difference of squares. It's a special type of sandwich in the algebra world.

Imagine you have two squares, one of them is the 'positive' square, and the other one is its 'negative' counterpart. The 'difference of squares' is just subtracting one from the other. To put it in math terms, it's like this formula: \(a^2 - b^2 = (a+b)(a-b)\). Now, let's say we have this math snack: \(x^{n}+2\) and \(x^{n}-2\). Use the difference of squares formula, and munch! We get \(x^{2n} - 4\), which simplifies our first piece of the puzzle just right.

Perfect Square Trinomials

Moving on to our second concept is like stepping into a bakery of squares, the 'perfect square trinomials.' These are trinomials that look all neat and powdered, just like our perfectly baked goods. The general recipe for creating them is \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), depending on whether you prefer a sweet or savory twist.

Back to our math bakery, let's whip up a batch with \(x^{n}\) as our main ingredient. Add a pinch of 3, and what do we have? A perfectly squared \(x^{n}-3\)^2 trinomial! Carefully mixing it gives us a delicious \(x^{2n} - 6x^n + 9\). That’s your second important piece of the algebraic expression, packed and ready to go.

Simplifying Algebraic Expressions

Once we have our freshly prepared pieces, the difference of squares and perfect square trinomials, it's time to combine them into one grand finale dish – simplifying algebraic expressions. Here's where we take all our ingredients and mix them up to create a simpler form.

Simplifying is basically reducing the expression, just like when you boil down a sauce to make it richer and more flavorful. You combine like terms, cancel what's unnecessary, and voilà, your algebraic expression has transformed! In our problem, we subtract the fluffy perfect square trinomial \(x^{2n} - 6x^n + 9\) from the crunchy difference of squares \(x^{2n} - 4\), following the order of operations. Don't forget to distribute the negative sign; a common slip-up is like forgetting the salt in your dish – it changes everything! You’ll end up with a simplified expression, looking all neat and tidy, \(6x^n - 13\).