Factorise each of the following 8a³-b³-12a²b+6ab²

Let's factor the expression \( 8a^3 - b^3 - 12a^2b + 6ab^2 \) step by step.

### Step 1: Group the terms
We can group the terms into two pairs:
\[
(8a^3 - 12a^2b) - (b^3 - 6ab^2)
\]

### Step 2: Factor out common factors from each group
From the first group \( 8a^3 - 12a^2b \), we can factor out \( 4a^2 \):
\[
4a^2(2a - 3b)
\]
From the second group \( b^3 - 6ab^2 \), we can factor out \( -b^2 \):
\[
-b^2(b - 6a)
\]

Thus, the expression becomes:
\[
4a^2(2a - 3b) - b^2(b - 6a)
\]

### Step 3: Rearrange the second term
Notice that \( b - 6a \) is similar to \( 6a - b \) but with opposite signs, so we can rewrite \( b - 6a = -(6a - b) \). Therefore:
\[
4a^2(2a - 3b) + b^2(6a - b)
\]

### Step 4: Factor by grouping
Now, both terms have the common factor \( (6a - b) \), so we can factor that out:
\[
(4a^2 + b^2)(6a - b)
\]

### Final answer:
The factorized form of \( 8a^3 - b^3 - 12a^2b + 6ab^2 \) is:
\[
(4a^2 + b^2)(6a - b)
\]

Would you like more details or further clarification on any part?

Here are some related questions to expand on this topic:
1. How do you factor the difference of cubes?
2. Can all cubic expressions be factorized by grouping?
3. What are the properties of polynomials that allow factorization by grouping?
4. How do you factor trinomials of higher degrees?
5. What is the role of the greatest common factor in factorization?

**Tip**: Always check for a greatest common factor (GCF) first before attempting more complex factorization techniques.

Learn how to factor the cubic expression 8a³-b³-12a²b+6ab² with a detailed step-by-step solution. This problem covers advanced algebraic concepts, including factoring by grouping and the difference of cubes theorem. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation