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Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $(2 - 3x)^5$ giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 3

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Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $(2 - 3x)^5$ giving each term in its simplest form.

Worked Solution & Example Answer:Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $(2 - 3x)^5$ giving each term in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 3

Step 1

Step 1: Identify the binomial expansion formula

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Answer

The binomial expansion for (a+b)n(a + b)^n can be expressed as: inom{n}{k} a^{n-k} b^k where inom{n}{k} is the binomial coefficient.

Step 2

Step 2: Apply the formula to $(2 - 3x)^5$

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Answer

For our case, let a=2a = 2, b=−3xb = -3x, and n=5n = 5. We will calculate the first three terms for k=0k = 0, 11, and 22.

  1. For k=0k = 0: inom{5}{0} (2)^{5} (-3x)^{0} = 1 imes 32 imes 1 = 32
  2. For k=1k = 1: inom{5}{1} (2)^{4} (-3x)^{1} = 5 imes 16 imes (-3x) = -240x
  3. For k=2k = 2: inom{5}{2} (2)^{3} (-3x)^{2} = 10 imes 8 imes 9x^2 = 720x^2

Step 3

Step 3: Combine the results

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Answer

The first three terms in ascending powers of xx are:

  • First term: 3232
  • Second term: −240x-240x
  • Third term: 720x2720x^2

Thus, the final result is: 32−240x+720x232 - 240x + 720x^2

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