Problem 108 - College Algebra

Question

Which one of the following is true? a. Because \(x^{2}+1\) is irreducible over the integers, it follows that \(x^{3}+1\) is also irreducible. b. One correct factored form for \(x^{2}-4 x+3\) is \(x(x-4)+3\) c. \(x^{3}-64=(x-4)^{3}\) d. None of the above is true.

Step-by-Step Solution

Answer

The correct answer is (d) None of the above is true. This is because none of the statements (a), (b), (c) turned out to be true during the evaluation process.

Step 1: Evaluate option (a)

Consider the equation \(x^{2}+1\). Yes, it is indeed irreducible over integers since it cannot be factored into integers. According to the statement, if \(x^{2}+1\) is irreducible over integers then \(x^{3}+1\) must also be irreducible. However, this is not the case because \(x^{3}+1\) can be factored over the integers into \((x+1)(x^{2}-x+1)\). Hence, option (a) is false.

Step 2: Evaluate option (b)

Factoring the equation \(x^{2}-4x+3\), it can be written as \((x-3)(x-1)\) and not \(x(x-4)+3\) as suggested. Hence, option (b) is also not true.

Step 3: Evaluate option (c)

Checking equation \(x^{3}-64\), it should equal to \((x-4)^{3}\) based on the given option. But \((x-4)^{3}\) expands to \(x^{3} - 12x^{2} + 48x - 64\) not \(x^{3}-64\), thus option (c) is false.

Key Concepts

Irreducible Polynomials

An irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two or more non-constant polynomials. The concept of irreducibility depends on the set of numbers we are allowed to use for the coefficients. For example, over the integers, the polynomial \(x^2 + 1\) is irreducible because there are no two polynomials with integer coefficients that multiply to give \(x^2 + 1\).

However, this property does not 'transfer' to higher powers automatically. For instance, \(x^3 + 1\) can indeed be factored into \((x + 1)(x^2 - x + 1)\). The confusion might arise due to the similarity in the apparent structure of the polynomials. Understanding this key difference is crucial because assuming irreducibility without proof can lead to incorrect conclusions in algebra.

Polynomial Factoring

The process of polynomial factoring involves breaking down a complex algebraic expression into simpler 'factor' polynomials whose product is equal to the original expression. Factoring is a fundamental tool in algebra as it simplifies expressions and solves equations. The factored form of \(x^2 - 4x + 3\) is \((x - 3)(x - 1)\), representing the roots where the expression equals zero.

When a polynomial is correctly factored, it allows us to understand its properties, such as its zeros (solutions) and behavior in a graph. Misinterpreting the factored form can lead to significant misunderstandings, as seen with option (b) from the exercise. Factoring polynomials often involves techniques such as finding the greatest common factor, using the difference of squares, or applying the quadratic formula.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetical operations (addition, subtraction, multiplication, division, and exponentiation by an exponent). It represents a particular numerical value when the variables are substituted by specific numbers. The art of manipulating these expressions involves understanding the basic operations, the role of parentheses, and the conventions for order of operations.

Each expression also has an 'expanded' and 'factored' form, which are equivalent in value but different in appearance. For instance, \((x - 4)^3\) is an expanded form and sometimes, due to misinterpretation, it might be incorrectly said to be equivalent to \(x^3 - 64\), which is not the case as the step-by-step solution illustrates. Recognizing algebraic expressions and manipulating them correctly is fundamental to solving a variety of problems in algebra.