Problem 134 - Algebra and Trigonometry

Question

Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. \(x^{4}-16\) is factored completely as \(\left(x^{2}+4\right)\left(x^{2}-4\right)\)

Step-by-Step Solution

Answer

The given expression is partially incorrect. The correct version considering complex numbers would be \(x^{4}-16 = (x^2 + 4i)((x + 2^{1/2})(x - 2^{1/2}))\), and without complex numbers, the expression \(x^{4}-16\) cannot be factored completely as given.

Step 1: Factorize the first square

The formula \(a^2 - b^2 = (a + b)(a - b)\) can be applied to the complete expression. In this case, \(a = x^2\) and \(b = (4)^{1/2}\). This gives: \(x^{4}-16 = (x^2 + 2^{1/2})(x^2 - 2^{1/2}).\)

Step 2: Factorize the second square

The second square can also be factored using the formula \(a^2 - b^2 = (a + b)(a - b)\). However, be mindful that it doesn't apply if b is a complex number. Here, \(a = x\) and \(b = (2)^{1/2}\). Thus, \(x^{2}-4 = ((x + 2^{1/2})(x - 2^{1/2}))\). However, it can result in a complex too.

Step 3: Compare the expressions and correct mistakes

Comparing the given expression \(\left(x^{2}+4\right)\left(x^{2}-4\right)\) with our factorized expressions, we can see that the first factor is not the same, it should be \(x^2 + 4i\) and not \(x^2 + 4\) if considering complex numbers, otherwise the expression cannot be totally factored as stated. The second factor \(x^2 - 4\) is factored correctly.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental concept in algebra. It involves expressing a polynomial as a product of simpler polynomials. For example, the expression \(x^4 - 16\) can be factored using specific algebraic identities. This makes solving equations or simplifying expressions much easier. One of the most useful identities for factoring is the difference of squares formula:

• \(a^2 - b^2 = (a + b)(a - b)\)

In our exercise, \(x^4 - 16\) appears as a difference of squares where \(a = x^2\) and \(b = 4\). By applying this identity, we can further break down the expression into \((x^2 + 4)(x^2 - 4)\). This process is not only handy in simplifying expressions but is also crucial for solving polynomial equations.

Difference of Squares

The difference of squares is a specific kind of factoring applicable when dealing with expressions of the form \(a^2 - b^2\). This identity simplifies the work of factoring such expressions because it directly breaks them into two binomial expressions \((a + b)(a - b)\).
For instance, in the problem \(x^4 - 16\), we first recognized \(x^4\) as \((x^2)^2\) and 16 as \(4^2\). These can be rewritten as a difference of squares: \((x^2)^2 - 4^2\), which can be factored using the identity into \((x^2 + 4)(x^2 - 4)\).
It is important to note that when the numbers involved are negative or complex, further factorization sometimes leads to complex numbers in the solutions. This is due to the roots taken during factorization which might result in complex numbers when resolving symbols like \(i\) for imaginary numbers.

Complex Numbers

Complex numbers are used when dealing with square roots of negative numbers. They are in the form \(a + bi\), where \(i\) is the imaginary unit, defined by \(i^2 = -1\). When factoring expressions with negative square terms, complex numbers often arise.
In our scenario, while factorizing \(x^2 + 4\), using complex numbers means expressing it as \((x + 2i)(x - 2i)\), because 4 is treated as \(2^2\) and considering \(+4\) can be expressed as \((2i)^2\).
This demonstrates how complex numbers can complete factorization processes that appear unfinished in a real-number context. Understanding and manipulating complex numbers enhances your capability to solve wider ranges of algebraic equations.

Algebraic Expressions

Algebraic expressions consist of terms formed by variables, coefficients, constant numbers, and arithmetic operations. Mastering these expressions involves understanding how they can be altered, simplified, or factored.
Factoring involves rewriting an expression as a product of its numbers or variables. In our exercise \(x^4 - 16\), this involved understanding the type of polynomial and using known algebraic identities to achieve a factored form.

• Polynomials like \(x^4\) and constants like 16 are both parts of these expressions.
• Recognizing patterns like the difference of squares reduces complexity.

Grasping how various algebraic components interact gives students the tools to simplify and understand more complex mathematical problems. This command over algebraic expressions forms the backbone for further exploration in mathematics, such as calculus and beyond.