Problem 41 - Algebra and Trigonometry

Question

39 \(-42\) . Factor using the Binomial Theorem. $$ 8 a^{3}+12 a^{2} b+6 a b^{2}+b^{3} $$

Step-by-Step Solution

Answer

The factored form is \( (2a + 3b)(4a^2 + b^2) \).

Step 1: Identify the polynomial

The given polynomial is \(8a^3 + 12a^2b + 6ab^2 + b^3\). Our goal is to factor this expression using the Binomial Theorem.

Step 2: Group terms

We start by grouping the terms into two groups for easier factoring. The polynomial can be rewritten as \((8a^3 + 12a^2b) + (6ab^2 + b^3)\).

Step 3: Factor each group separately

Factor through common factors in each group. For the first group \(8a^3 + 12a^2b\), factor out the common factor \(4a^2\), which gives us \(4a^2(2a + 3b)\). For the second group \(6ab^2 + b^3\), factor out \(b^2\), resulting in \(b^2(6a + b)\).

Step 4: Factor the expression completely

We observe that \(6a + b\) can be written as \(2a + 3b\). Hence, rewrite the grouped factors to show a common binomial factor. Our grouped expression is \(4a^2(2a + 3b) + b^2(2a + 3b)\).

Step 5: Factor out the common binomial factor

Now, factor out the common binomial \((2a + 3b)\) from the expression: \((2a + 3b)(4a^2 + b^2)\). This factorization reflects the use of the binomial theorem.

Key Concepts

Polynomial Factorization

Polynomial factorization is a process used to break down a polynomial into simpler terms that multiply together to give the original polynomial. It is like taking a number, such as 12, and expressing it as a product of its factors, like 2 and 6. Factoring polynomials can simplify the expression or make solving equations easier.

In the given exercise, we start with the polynomial \(8a^3 + 12a^2b + 6ab^2 + b^3 \). Our goal is to factor it into a product of simpler expressions. Proper factorization hinges on recognizing patterns and common factors, which help in breaking down the complex polynomial into manageable pieces. This process often involves rearranging terms, dividing, and extracting common factors from parts of the polynomial, ultimately simplifying the entire equation. Understanding how to manipulate these terms is essential in mathematics, especially when working with higher-degree polynomials.

Common Binomial Factor

A common binomial factor is an expression that appears as a factor in multiple parts of the polynomial. By identifying and extracting this common factor, we can simplify the polynomial into a product.

In our example, after grouping the expression \((8a^3 + 12a^2b) + (6ab^2 + b^3)\), we noticed that both groups could be expressed using a similar binomial, \((2a + 3b)\). This commonality allows us to rewrite the expression to clearly show the shared factor: \(4a^2(2a + 3b) + b^2(2a + 3b)\).

By factoring out the common binomial \((2a + 3b)\), we efficiently simplify the polynomial to \((2a + 3b)(4a^2 + b^2)\). Finding a common binomial factor is one of the key lessons in polynomial factorization, as it helps reduce complex expressions into more tractable forms.

Factor by Grouping

Factoring by grouping is a technique used when a polynomial has terms that can be paired into groups that share a common factor. This method helps simplify the expression by breaking it into smaller parts that are easier to manage.

In the provided solution, the polynomial \(8a^3 + 12a^2b + 6ab^2 + b^3\) is grouped into two parts: \((8a^3 + 12a^2b)\) and \((6ab^2 + b^3)\). Each group is considered separately to find common factors:

• For \(8a^3 + 12a^2b\), the common factor is \(4a^2\), giving us \(4a^2(2a + 3b)\).
• For \(6ab^2 + b^3\), the common factor is \(b^2\), resulting in \(b^2(6a + b)\).

By strategically rearranging and breaking down the polynomial, we managed to reveal the common factor \((2a + 3b)\) and ultimately simplify the expression. This method is particularly helpful for polynomials where terms naturally form pairs, emphasizing the importance of recognizing patterns in algebraic expressions.