Problem 67 - 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. $$9 y^{2}+64$$

Step-by-Step Solution

Answer

The polynomial \(9y^{2}+64\) is fully factorized to \((3y+8i)(3y-8i)\) which is a factorization over complex numbers, therefore it is prime over real numbers

Step 1: Identify the Type of Polynomial

The given polynomial is \(9y^{2}+64\). This polynomial can be seen as \(9y^{2} + 8^{2}\), which is a sum of squares. Therefore, the correct factoring formula for \(a^{2} + b^{2}\) should be applied here.

Step 2: Apply the Factoring Formula

The factoring formula for a sum of squares, \(a^{2} + b^{2}\), results in \((a+bi)(a-bi)\) which implies that it cannot be factored over the set of real numbers but over complex numbers. Substituting a = 3y and b = 8 into the formula yields \((3y+8i)(3y-8i)\)

Step 3: Check Factorization

Now multiply the two factors \((3y+8i)(3y-8i)\) and verify that the result is identical to the original expression. The expansion of this product is \(9y^{2} - (8i)^{2}\) which simplifies to \(9y^{2} + 64\), thus confirming that the factorization is valid.

Key Concepts

Understanding the Sum of Squares

The sum of squares is a concept found in algebra where you work with expressions of the form \(a^2 + b^2\). At first glance, unlike the difference of squares \(a^2 - b^2\), it may seem that the sum of squares cannot be factored using integers or real numbers. However, they become factorable when you include complex numbers. This is because the sum of squares will not yield real number roots as there is no real number solution for negative square roots. Let's take a closer look at an example. If you consider \(9y^2 + 64\), you can view this as \( (3y)^2 + 8^2\). Here, both terms are perfect squares. Understanding that you can rewrite the expression using squares is the first step in breaking it down into factors. Recognizing these patterns helps in identifying when to use complex numbers in further steps.

Introduction to Complex Numbers in Polynomial Factorization

Complex numbers expand the toolkit of factoring beyond the real number system. A complex number is of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part (with \(i\) being the imaginary unit, defined as \(i = \sqrt{-1}\)). Complex numbers allow us to solve equations that do not have real solutions.In terms of polynomial factorization, complex numbers come into play when dealing with the sum of squares. Using the example of \(9y^2 + 64\), we can factor it using complex numbers as \((3y + 8i)(3y - 8i)\). Notice how the imaginary unit \(i\) is essential here, making it possible to completely factor the polynomial. This approach is particularly useful because it confirms that while the polynomial doesn't have real factors, it can be expressed through its complex roots.

Employing Factoring Techniques with Polynomials

Factoring techniques are strategies we use to break down polynomials into simpler, multipliable components. The prime goal is to express the polynomial as a product of its factors. Basic techniques include identifying common factors, recognizing patterns like the difference of squares or perfect square trinomials, and factoring by grouping. However, with more complex polynomials, you might need to apply multiple techniques and even venture into complex numbers.In our specific example of the polynomial \(9y^2 + 64\), identifying it as a sum of squares is the starting point. Once you know it’s a sum of squares, and recognize that real factors don't exist, you shift to using complex number techniques. By employing the sum of squares in the complex number field, we could proceed to generate its factors: \((3y + 8i)(3y - 8i)\). Checking the factorization through multiplication confirms its validity, verifying it aligns accurately with the original polynomial. These advanced factoring techniques are imperative for solving complex polynomials, especially when real solutions are unattainable.