Problem 47 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$x^{2}+64$$

Step-by-Step Solution

Answer

A complete factorization over the complex numbers for the polynomial \(x^{2}+64\) is \((x-8i)(x+8i)\).

Step 1: Write Down the Given Polynomial

The given polynomial is \(x^{2}+64\).

Step 2: Identifying a and b

Since the given polynomial is a sum of squares, it can be written in the form of \(a^{2}+b^{2}\). Here \(a = x\) and \(b =8\) (as \(64 = 8^{2}\)).

Step 3: Apply the Identity

Then, apply the identity for the sum of squares, \(a^{2}+b^{2} = (a-ib)(a+ib)\), to perform the factorization, which gives us \((x-8i)(x+8i)\).

Key Concepts

Sum of Squares

When we come across the term 'sum of squares', it usually refers to the specific form of an expression where two terms are squared and added together, mathematically presented as \( a^{2} + b^{2} \). This structure is particularly interesting in algebra because, unlike the 'difference of squares' which can be factored into \(a+b)(a-b)\), the 'sum of squares' does not factor over the real numbers. This is because no two real numbers can square and sum up to a negative result which is necessary for real number factorization.

However, the 'sum of squares' can be factored over the complex numbers using the identity \( a^{2} + b^{2} = (a + bi)(a - bi)\), where \(i\) is the imaginary unit. This is very useful in advanced mathematics, particularly when dealing with polynomial factorization where one might encounter quadratic expressions that are not easily factorable using real numbers only.

Complex Numbers

In the world of algebra, the concept of complex numbers is a fundamental one. A complex number is composed of a real part and an imaginary part, typically expressed as \( a + bi\), where \(a \) and \( b \) are real numbers, and \( i \) is the square root of -1.

The imaginary unit \(i\) is a mathematical creation that extends the real number system to include solutions to equations that have no solution within the set of real numbers, such as the square root of a negative number. They are essential for solving polynomial equations that do not have real solutions. As in our exercise, where we encounter the expression \(x^{2}+64\), complex numbers allow us to factor this sum of squares by recognizing it as \(x^{2} + (8i)^{2}\) and then applying appropriate identities.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They are the backbone of algebra and make it easier to manipulate and factorize algebraic expressions. A familiar algebraic identity is the 'difference of squares', which allows us to factor expressions like \( a^{2} - b^{2} \) into \( (a+b)(a-b) \).

While solving the given exercise, we use a less commonly applied identity for the 'sum of squares'. Unlike most algebraic identities used for factoring, the 'sum of squares' does not have a real number factorization but rather a complex one, \( a^{2} + b^{2} = (a + bi)(a - bi)\). This identity is especially useful as it provides us with a method of factorization when working with polynomials that do not factor nicely with real numbers alone.