Problem 56 - Algebra 2

Question

CHALLENGE. Factor \(64 p^{2 n}+16 p^{n}+1\)

Step-by-Step Solution

Answer

The expression factors as \((8p^n + 1)^2\).

Step 1: Identify Factorable Form

Observe that the given expression \(64 p^{2n} + 16 p^n + 1\) resembles a perfect square trinomial. These trinomials typically have the form \(a^2 + 2ab + b^2\). Notice that \((8p^n)^2 = 64p^{2n}\) and \(1^2 = 1\).

Step 2: Attempt Factoring as a Perfect Square

Let's compare the given expression with the expanded form \((a + b)^2 = a^2 + 2ab + b^2\). Here, consider \(a = 8p^n\) and \(b = 1\). Calculate \(2ab = 2 \times 8p^n \times 1 = 16p^n\), which matches the middle term of our trinomial.

Step 3: Write the Perfect Square Form

Since the expression can be written as a perfect square trinomial, we have \((8p^n + 1)^2 = 64p^{2n} + 16p^n + 1\). Thus, the factorization is \((8p^n + 1)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial. It has three terms and can be expressed in the form \(a^2 + 2ab + b^2\). This structure allows the trinomial to be rewritten as \((a + b)^2\). Recognizing the perfect square form is crucial for factoring efficiently, as it simplifies the expression into a product of identical binomials.

To determine if a trinomial is a perfect square:

• Check if the first and last terms are perfect squares themselves.
• Calculate the square root of each and ensure the middle term equals \(2ab\).

In our example, \(64p^{2n}\) is \((8p^n)^2\) and 1 is \(1^2\). Comparing, we verify the middle term \(16p^n\) matches after calculation as \(2 \times 8p^n \times 1\). Recognizing this pattern simplifies the process of expressing the trinomial as \((8p^n + 1)^2\).

Factoring Techniques

Factoring is a crucial skill in algebra that decomposes expressions into products of simpler factors. This process simplifies solving equations and evaluating algebraic expressions efficiently. There are several methods for factoring:

• **Greatest Common Factor (GCF):** Simplify by removing the highest common factor across terms.
• **Grouping:** Pairing terms to identify and extract common factors from each pair.
• **Difference of Squares:** Recognize expressions of the form \(a^2 - b^2 = (a+b)(a-b)\).
• **Perfect Square Trinomials:** As discussed, these take the form \(a^2 + 2ab + b^2\).

By applying these techniques, you can break down complex expressions like \(64p^{2n} + 16p^n + 1\) into simpler products. Identifying the correct technique, such as recognizing a perfect square trinomial, helps ensure efficient solutions.

Algebraic Expressions

Engaging with algebraic expressions involves a variety of operations and transformations. These expressions involve variables, numbers, and arithmetic operations. Mastery involves not just arithmetic skills, but also an understanding of underlying patterns and structures.

When tackling algebraic expressions, be intuitive with:

• **Polynomials:** Combinations of terms involving powers of variables, like \(64p^{2n}+16p^{n}+1\).
• **Simplification:** Reducing expressions by combining like terms and utilizing distributive properties.
• **Factoring:** Rewriting expressions as products of simpler terms while retaining equality.

Understanding these aspects provides a foundation for solving equations, graphing, and interpreting mathematical models. Working through examples like the given polynomial allows for recognizing specific structures, aiding in efficient problem-solving and comprehension.