Math Problem Statement

Expand the given binomial: (4x - 3y)^5

Solution

The binomial in your image is (4x3y)5(4x - 3y)^5. To expand this, we will use the binomial theorem, which states:

(a+b)n=k=0n(nk)ankbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Here:

  • a=4xa = 4x,
  • b=3yb = -3y,
  • n=5n = 5.

The expansion will involve the binomial coefficients (5k)\binom{5}{k} for k=0,1,2,3,4,5k = 0, 1, 2, 3, 4, 5, where each term follows the pattern:

(5k)(4x)5k(3y)k\binom{5}{k} (4x)^{5-k} (-3y)^k

Let's calculate the terms one by one:

  1. For k=0k=0: (50)(4x)5(3y)0=1(4x)5=1024x5\binom{5}{0} (4x)^5 (-3y)^0 = 1 \cdot (4x)^5 = 1024x^5
  2. For k=1k=1: (51)(4x)4(3y)1=5256x4(3y)=3840x4y\binom{5}{1} (4x)^4 (-3y)^1 = 5 \cdot 256x^4 \cdot (-3y) = -3840x^4y
  3. For k=2k=2: (52)(4x)3(3y)2=1064x39y2=5760x3y2\binom{5}{2} (4x)^3 (-3y)^2 = 10 \cdot 64x^3 \cdot 9y^2 = 5760x^3y^2
  4. For k=3k=3: (53)(4x)2(3y)3=1016x2(27y3)=4320x2y3\binom{5}{3} (4x)^2 (-3y)^3 = 10 \cdot 16x^2 \cdot (-27y^3) = -4320x^2y^3
  5. For k=4k=4: (54)(4x)1(3y)4=54x81y4=1620xy4\binom{5}{4} (4x)^1 (-3y)^4 = 5 \cdot 4x \cdot 81y^4 = 1620xy^4
  6. For k=5k=5: (55)(4x)0(3y)5=1(243y5)=243y5\binom{5}{5} (4x)^0 (-3y)^5 = 1 \cdot (-243y^5) = -243y^5

So, the expanded form is:

(4x3y)5=1024x53840x4y+5760x3y24320x2y3+1620xy4243y5(4x - 3y)^5 = 1024x^5 - 3840x^4y + 5760x^3y^2 - 4320x^2y^3 + 1620xy^4 - 243y^5

Would you like further explanation on any step?

Here are some related questions to expand on this:

  1. How does the binomial theorem apply to higher powers?
  2. How do binomial coefficients relate to Pascal's Triangle?
  3. What is the general form of a binomial expansion?
  4. How does expanding a binomial differ for positive and negative terms?
  5. What are some real-world applications of the binomial expansion?

Tip: When expanding binomials, ensure the correct application of negative signs to avoid errors, especially when powers of negative numbers are involved.

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Math Problem Analysis

Mathematical Concepts

Algebra
Binomial Theorem
Polynomials

Formulas

(a + b)^n = sum_{k=0}^{n} binom(n,k) a^{n-k} b^k

Theorems

Binomial Theorem

Suitable Grade Level

Grades 10-12