Math Problem Statement

(a-2b)^3 answer first then soulution, then what method should I use? FOIL or box?

Solution

Answer:

The expression (a2b)3(a - 2b)^3 expands to a36a2b+12ab28b3a^3 - 6a^2b + 12ab^2 - 8b^3.

Solution:

To expand (a2b)3(a - 2b)^3, we can follow these steps:

  1. Start with the Binomial Theorem: The Binomial Theorem states that (x+y)n(x + y)^n can be expanded as: k=0n(nk)xnkyk\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k Here, x=ax = a, y=2by = -2b, and n=3n = 3.

  2. Apply the Binomial Expansion: Expand (a2b)3(a - 2b)^3 using the binomial coefficients: (a2b)3=(30)a3(2b)0+(31)a2(2b)1+(32)a1(2b)2+(33)a0(2b)3(a - 2b)^3 = \binom{3}{0}a^3(-2b)^0 + \binom{3}{1}a^2(-2b)^1 + \binom{3}{2}a^1(-2b)^2 + \binom{3}{3}a^0(-2b)^3 Simplifying each term:

    • (30)a3(2b)0=1a31=a3\binom{3}{0}a^3(-2b)^0 = 1 \cdot a^3 \cdot 1 = a^3
    • (31)a2(2b)1=3a2(2b)=6a2b\binom{3}{1}a^2(-2b)^1 = 3 \cdot a^2 \cdot (-2b) = -6a^2b
    • (32)a(2b)2=3a4b2=12ab2\binom{3}{2}a(-2b)^2 = 3 \cdot a \cdot 4b^2 = 12ab^2
    • (33)(2b)3=1(8b3)=8b3\binom{3}{3}(-2b)^3 = 1 \cdot (-8b^3) = -8b^3
  3. Combine the terms: Putting it all together gives: (a2b)3=a36a2b+12ab28b3(a - 2b)^3 = a^3 - 6a^2b + 12ab^2 - 8b^3

Method: FOIL or Box?

For this specific problem:

  • Neither FOIL nor the box method is typically used directly to expand a binomial raised to a power greater than 2.
  • Binomial expansion is more appropriate.

However, if you are expanding a cubic binomial like this by multiplying step-by-step, then Box method (also known as area method) can be useful for organizing the multiplication in stages. For example, first multiply (a2b)(a - 2b) by (a2b)(a - 2b), then multiply the result by (a2b)(a - 2b) again.

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

Follow-Up Questions:

  1. How would you expand (a+3b)3(a + 3b)^3 using the Binomial Theorem?
  2. Can you use the Box method to expand (a2b)2(a - 2b)^2?
  3. What are the advantages of using the Binomial Theorem over FOIL for higher powers?
  4. How does the sign of the middle terms change when expanding (a+b)3(a + b)^3 compared to (ab)3(a - b)^3?
  5. What is the difference between the Binomial Theorem and Pascal's Triangle in expansion?

Tip:

When expanding binomials, remember that the Binomial Theorem works efficiently for any power, especially when the exponent is greater than 2.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Binomial Theorem
Binomial Expansion

Formulas

Binomial Theorem: (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Theorems

Binomial Theorem

Suitable Grade Level

High School