Problem 38 - Elementary Algebra

Question

Factor completely. $$ (x 2+2) 2-(x-1) 2 $$

Step-by-Step Solution

Answer

\((x^2 - x + 3)(x^2 + x + 1)\)

Step 1: Identify the Structure of the Expression

Recognize that the given expression \( (x^2+2)^2 - (x-1)^2 \) is of the form \( a^2 - b^2 \), which is a difference of squares.

Step 2: Apply the Difference of Squares Formula

Use the formula \( a^2 - b^2 = (a-b)(a+b) \) on the expression, where \( a = (x^2+2) \) and \( b = (x-1) \). This gives:\[(x^2 + 2 - (x - 1))(x^2 + 2 + (x - 1))\]

Step 3: Simplify the Terms

Simplify each term in the factored expression:- First term: \( x^2 + 2 - x + 1 = x^2 - x + 3 \)- Second term: \( x^2 + 2 + x - 1 = x^2 + x + 1 \)

Step 4: Write the Final Factorized Expression

Combine the simplified terms to write the fully factorized form:\[(x^2 - x + 3)(x^2 + x + 1)\]

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that helps in factoring certain types of polynomial expressions. If you have an expression in the form of \( a^2 - b^2 \), it can be rewritten as \((a - b)(a + b)\). This formula takes advantage of the fact that when you subtract one square number from another, you can split it into the product of two simpler expressions.

Let's take a closer look at the math behind it: when expanded, \((a - b)(a + b)\) simplifies to \(a^2 - b^2\) because the middle terms \(ab\) and \(-ab\) cancel each other out. This identity is particularly useful because it simplifies otherwise complex equations and makes it easier to find their roots or further factor them if needed.

In exercises, like the one you've completed, understanding the difference of squares allows you to break down what initially looks like a complicated expression into simpler parts. The key is to identify that specific \(a^2 - b^2\) structure and then apply the formula accordingly.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables raised to various powers and multiplied by coefficients. In general terms, a polynomial looks like: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where each \(a\) represents a coefficient, and \(x\) is a variable with non-negative integer exponents.

Polynomials can have different degrees based on the highest power of the variable present in the expression. For example, the polynomial \(x^2 + 5x + 6\) is a quadratic polynomial because the highest power of \(x\) is 2.

When factoring polynomial expressions, the goal is to break them down into simpler terms, typically multiplying two binomials or polynomials of lower degrees. Factoring helps to reveal the roots and simplify complex expressions for easier computation and analysis, which is what you did with your exercise. Recognizing polynomial structures and the rules around them is crucial for algebra proficiency.

Algebraic Identities

Algebraic identities are pre-established expressions or formulas that are universally true for all values of the involved variables. They provide mathematicians and students a shortcut to solve equations more easily. Some common algebraic identities include the square of a sum \((a + b)^2 = a^2 + 2ab + b^2\), the square of a difference \((a - b)^2 = a^2 - 2ab + b^2\), and the difference of squares \(a^2 - b^2 = (a-b)(a+b)\).

These identities are fundamental because they allow us to manipulate and solve algebraic expressions efficiently, saving the time and complexity of expanding expressions every time. Additionally, they are used in deriving more complex formulas and can serve as the basis for proving mathematical theorems.

In your exercise, realizing the expression was a specific algebraic identity allowed you to instantly apply a known fact—unveiling a factored form directly, instead of through lengthy computations. Think of these identities as powerful tools in your mathematical toolkit, ready to bring clarity to complicated problems.