Problem 89 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 n^{2}+64$$

Step-by-Step Solution

Answer

The polynomial \(25n^2 + 64\) is not factorable over the integers.

Step 1: Identify the Form of the Polynomial

Examine the polynomial given: \(25n^2 + 64\). Notice that this is not a difference of squares, trinomial, or a common factor scenario. It is a sum of squares which generally cannot be factored using integers.

Step 2: Consider Special Factoring Techniques

Sums of squares do not factor over the integers in the way that differences of squares do. Check if it is an exception or can be factored using complex numbers or clear the special cases (like \((a+b)^2\) or \((a-b)^2\)) to confirm.

Step 3: Conclude Factorability Using Integers

Since \(25n^2 + 64\) doesn't match any common factorizable form or any special identity that will allow factoring over the integers, it is confirmed that the polynomial is not factorable using integers.

Key Concepts

sum of squares

The **sum of squares** is a concept that often trips up many students. When you see an expression like \(a^2 + b^2\), it can be tempting to try to factor it. However, unlike the difference of squares, the sum of squares typically cannot be factored using integers.
For instance, take the expression \(25n^2 + 64\). Here, both terms \(25n^2\) and \(64\) are perfect squares. The term \(25n^2\) is \((5n)^2\) and \(64\) is \(8^2\). Despite being squares, this does not open the door to factoring with integers because of the addition.
A special case occurs when factoring over complex numbers where sums of squares can be expressed using imaginary numbers, but in the typical high school context involving integers, there isn't a clean factorization.

difference of squares

The **difference of squares** is a factoring technique that is considerably more familiar and useful in algebra. It applies to expressions that are reduced to the form \(a^2 - b^2\). These can always be factored into \((a - b)(a + b)\).
For example, if you have an expression like \(49n^2 - 64\), you can recognize this as a difference of squares because both 49\(n^2\) and 64 are perfect squares.

• \(49n^2 = (7n)^2\)
• \(64 = 8^2\)

So, \(49n^2 - 64\) factors neatly into \((7n - 8)(7n + 8)\).
Recognizing a difference of squares can often be the key to simplifying more complex algebraic expressions. It is a straightforward, yet powerful tool.

integer factorability

**Integer factorability** refers to our ability to factor polynomials into expressions whose coefficients are integers. This concept is foundational in algebra because it helps simplify expressions and solve equations more easily.
Consider the exercise with \(25n^2 + 64\). As analyzed, this polynomial is a sum of squares, which means it is not factorable using integers. It tells us that sometimes, polynomials cannot be simplified into products of smaller degree polynomials with integer coefficients.
Being aware of integer factorability also aids in identifying when a polynomial remains in its simplest form, such as a sum of squares. Identifying when polynomial expressions are factorizable can save time and prevent errors.

special factoring techniques

Learning about **special factoring techniques** is crucial as they provide various ways to break down complex polynomials. Some of these special techniques include recognizing forms like the difference of squares or special products like \((a + b)^2\) and \((a - b)^2\).
These techniques often involve identifying patterns within the expression that match known formulas.

• The difference of squares, for example, always breaks down into \((a - b)(a + b)\)
• The perfect square trinomial \((a + b)^2\) expands to \(a^2 + 2ab + b^2\)

By knowing these techniques, you can approach problems strategically to find solutions effectively.
In the case of the sum of squares, awareness of these techniques can lead you to understand why it doesn’t fit these patterns, thus confirming a lack of integer factorability. These strategies are key tools in an algebraist's toolkit.