Math Problem Statement

Factorise a to to the power 3 b minus AP to the power 3 using middle term splitting solve

Solution

Let's factorize the expression a3bab3a^3b - ab^3 using the middle term splitting method. The expression is:

a3bab3a^3b - ab^3

Step 1: Factor out the common terms

We can see that both terms have a common factor of abab. Let's factor it out first:

a3bab3=ab(a2b2)a^3b - ab^3 = ab(a^2 - b^2)

Step 2: Apply the difference of squares

Now, we are left with the expression a2b2a^2 - b^2, which is a difference of squares. We can factor it as:

a2b2=(ab)(a+b)a^2 - b^2 = (a - b)(a + b)

Step 3: Write the final factorized form

Substitute the factorization of a2b2a^2 - b^2 back into the expression:

a3bab3=ab(ab)(a+b)a^3b - ab^3 = ab(a - b)(a + b)

Thus, the fully factorized form of a3bab3a^3b - ab^3 is:

ab(ab)(a+b)ab(a - b)(a + b)

Would you like further details on any of these steps or have any specific questions?

Here are 5 related questions:

  1. How would you factor other cubic expressions, like a3b3a^3 - b^3?
  2. Can you apply the middle term splitting method to polynomials of higher degrees?
  3. What is the difference of squares, and how can it be used in other factorizations?
  4. How do you recognize common factors in complex algebraic expressions?
  5. What are other factorization techniques besides middle term splitting?

Tip: Always look for common factors first when attempting to factor any expression—it simplifies the process significantly.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factorization
Difference of Squares

Formulas

a^3b - ab^3 = ab(a^2 - b^2)
a^2 - b^2 = (a - b)(a + b)

Theorems

Difference of Squares Theorem

Suitable Grade Level

Grades 9-10