Problem 33 - Algebra 2 and Trigonometry

Question

In \(27-39,\) factor each polynomial completely. $$ x^{4}-16 $$

Step-by-Step Solution

Answer

The complete factorization is \((x^2 + 4)(x+2)(x-2)\).

Step 1: Recognize the Formula

The given polynomial is of the form \( x^4 - 16 \). Notice that this is a difference of squares because it can be rewritten as \( (x^2)^2 - 4^2 \). The difference of squares formula is \( a^2 - b^2 = (a+b)(a-b) \).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula to \( (x^2)^2 - 4^2 \). Set \( a = x^2 \) and \( b = 4 \), so the expression becomes \( (x^2 + 4)(x^2 - 4) \).

Step 3: Recognize the New Difference of Squares

The second term \( x^2 - 4 \) is also a difference of squares. It can be factorized using the formula again, since \( x^2 - 4 = (x^2 - 2^2) \).

Step 4: Apply the Formula Again

Factor \( x^2 - 4 \) as \( (x+2)(x-2) \) using the difference of squares formula.

Step 5: Combine All Factors

Combine all the factors from the previous steps: \((x^2 + 4)(x+2)(x-2)\). This is the complete factorization of the original polynomial.

Key Concepts

Difference of Squares

The concept of the difference of squares is an essential tool in polynomial factorization. It refers to expressions that can be written in the form \( a^2 - b^2 \), where both \( a \) and \( b \) are perfect squares. The formula for factoring a difference of squares is \( a^2 - b^2 = (a + b)(a - b) \). This pattern allows us to simplify and solve a wide range of algebraic expressions.

When you see a polynomial like \( x^4 - 16 \), you can recognize it as a difference of squares because it can be rewritten as \((x^2)^2 - 4^2\). To factor it, identify \( a \) as \( x^2 \) and \( b \) as \( 4 \). Then, apply the formula to split it into \((x^2 + 4)(x^2 - 4)\). Note that breaking down such expressions using the difference of squares formula is a powerful skill in algebra.

Factorization

Factorization is the process of breaking down a complex expression into simpler factors that, when multiplied together, return the original expression. This is a key concept in simplifying algebraic equations and solving polynomial expressions.

When faced with the polynomial \( x^4 - 16 \), understanding how to factor it completely is crucial. First, recognize the difference of squares and factor it into \((x^2 + 4)(x^2 - 4)\). This step simplifies the expression by reducing the powers of the terms involved.

The goal is always to express the polynomial as a product of irreducible factors. In this case, after applying the difference of squares once, we find that \( (x^2 - 4) \) is itself a difference of squares. This means we can factor it further into \((x+2)(x-2)\). The complete factorization of \( x^4 - 16 \) becomes \((x^2 + 4)(x + 2)(x - 2)\).

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is the building block for understanding deeper mathematical concepts and solving equations. Learning how to recognize patterns such as the difference of squares is part of honing your skills in algebra.

The factorization of a polynomial like \( x^4 - 16 \) relies heavily on algebraic techniques and formulas. Algebra gives us the tools to rewrite mathematical expressions in ways that make them easier to work with. By mastering the difference of squares, students can quickly break down complex expressions, making them manageable and simple to solve.

• Recognizing patterns: The ability to see \( (x^2)^2 - 4^2 \) as a difference of squares is a fundamental skill.
• Simplifying expressions: Using algebraic rules allows you to reduce an expression to its simplest form.
• Solving problems: Algebraic techniques make it easier to solve equations, find roots, and work through polynomial factorization.

Polynomials

Polynomials are expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding polynomials and how to manipulate them is a critical skill in algebra.

The expression \( x^4 - 16 \) is a polynomial and, like all polynomials, can be broken down or factored. Recognizing that this specific polynomial is a difference of squares helps in immediately identifying a method to simplify it.

Working with polynomials requires being comfortable with the processes of factorization and expansion, and the ability to identify special forms like the difference of squares. Noticing these special forms allows you to efficiently break down polynomials into simpler components:

• Understanding structure: Polynomials can be thought of as building blocks in mathematical expressions.
• Expanding and factoring: Techniques for expanding and factoring polynomials are fundamental tools in algebra.
• Simplifying calculations: By breaking down polynomials, you simplify calculations, making the problem easier to handle.