Problem 77 - College Algebra

Question

Factor completely, or state that the polynomial is prime. $$ x^{2}+64 $$

Step-by-Step Solution

Answer

\(x^{2}+64\) can be completely factored into \((x+8i)(x-8i)\).

Step 1: Recognize the Binomial Form

The existing expression \(x^{2}+64\) is a binomial of the form \(a^{2}+b^{2}\), where \(a = x\) and \(b=8\) (since \(8^2 = 64\)).

Step 2: Apply the Sum of Squares Formula

The standard form for the sum of squares is \(a^{2} + b^{2} =(a + b \sqrt{-1}) (a - b \sqrt{-1})\). Substituting \(a = x\) and \(b = 8\) into the sum of squares formula we get, \(x^{2} + 64 = (x+8i)(x-8i)\) where \(i = \sqrt{-1}\).

Key Concepts

Binomial Expressions

Binomial expressions are polynomial expressions that involve exactly two terms. A common representation of a binomial is of the form \( a^2 + b^2 \), where both \( a \) and \( b \) can each be any expression with variables. Binomials can be tricky when it comes to factoring, especially when they are sums of squares, as not all can be easily factored using real numbers.
When you're factoring binomials, you need to be aware of various forms and identities, such as the difference of squares, which is \( a^2 - b^2 = (a + b)(a - b) \), or the special case of perfect square trinomials. However, recognizing the specific form of the binomial is key to using the correct factoring approach. For example, the sum \( a^2 + b^2 \) is often mistakenly identified as prime (not factorable), but when extended to complex numbers, it can be factored.

Sum of Squares Formula

The sum of squares formula deals with expressions of the form \( a^2 + b^2 \). Unlike the difference of squares, which presents a straightforward factorization involving real numbers, the sum of squares requires complex numbers to be factored To factor \( a^2 + b^2 \), you use the identity: \( a^2 + b^2 = (a + bi)(a - bi) \), where \( i \) is the imaginary unit \( \sqrt{-1} \).
This may sound complicated if you are new to complex numbers, but it's simply an extension of number systems that allows these kinds of expressions to be broken down further. For example, the expression \( x^2 + 64 \) fits the \( a^2 + b^2 \) format where \( a = x \) and \( b = 8 \). Using the sum of squares formula, it becomes \((x + 8i)(x - 8i)\).
Using this formula helps to factor expressions which otherwise seem prime when only considering real numbers. It simplifies the expression by representing it as a product, providing further insights into its structure.

Complex Numbers

Complex numbers may initially seem challenging, but they're not as complex as they sound. A complex number is typically written in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
The real part of the complex number is \( a \), and the imaginary part is \( b \). Complex numbers come in handy particularly in solving equations that cannot be solved within the set of real numbers. They expand the number system we can use to perform calculations and solve problems.
When factoring polynomials like \( x^2 + 64 \), complex numbers are essential. We use them to express the factored form as \((x + 8i)(x - 8i)\), showing that while it may be simple in appearance, employing complex numbers is necessary to fully understand its composition. This allows us to extend our ability to factor beyond the limitations imposed by real numbers.