Using binomial theorem to expand (2-¼x)⁴
(a - b)^n = Sum of all [(-1)^r]*(nCr)*(a^r)*[b^(n-r)]
Where 'r' varies from 0 to n.
a = 2
b = (1/4)x
n = 4
r = 0,1,2,3,4
Plug in the values, sum it up.
To expand the expression (2 - ¼x)⁴ using the binomial theorem, follow these steps:
Step 1: Determine the value of "n".
In this case, the expression is raised to the 4th power, so "n" is equal to 4.
Step 2: Identify the terms "a" and "b".
The expression (2 - ¼x) can be divided into two terms:
a = 2
b = -¼x
Step 3: Apply the binomial theorem formula.
The binomial theorem formula states that for any positive integer "n", the expansion of (a + b)ⁿ is given by:
(a + b)ⁿ = C(n, 0)aⁿb⁰ + C(n, 1)aⁿ⁻¹b¹ + C(n, 2)aⁿ⁻²b² + ... + C(n, n-1)abⁿ⁻¹ + C(n, n)a⁰bⁿ
Where C(n, r) is the binomial coefficient, calculated using the formula:
C(n, r) = n! / (r!(n - r)!)
Step 4: Calculate each term of the expansion.
Using the formula, we can expand (2 - ¼x)⁴ as follows:
(2 - ¼x)⁴ = C(4, 0)2⁴(-¼x)⁰ + C(4, 1)2³(-¼x)¹ + C(4, 2)2²(-¼x)² + C(4, 3)2¹(-¼x)³ + C(4, 4)2⁰(-¼x)⁴
Step 5: Simplify the expression.
Now we can simplify each term:
C(4, 0)2⁴(-¼x)⁰ = 1 * 2⁴ * (-¼x)⁰ = 2⁴ = 16
C(4, 1)2³(-¼x)¹ = 4 * 2³ * (-¼x) = -8x
C(4, 2)2²(-¼x)² = 6 * 2² * (¼x)² = 6 * 4 * (¼x)² = 6 * 4 * (1/16) * x² = 3x²
C(4, 3)2¹(-¼x)³ = 4 * 2¹ * (-¼x)³ = 4 * 2 * (-1/64) * x³ = -x³/8
C(4, 4)2⁰(-¼x)⁴ = 1 * 2⁰ * (-¼x)⁴ = (1/256) * x⁴ = x⁴/256
Step 6: Combine the terms.
Combine all the terms to get the final expanded form:
(2 - ¼x)⁴ = 16 - 8x + 3x² - x³/8 + x⁴/256
Therefore, the expanded form of the expression (2 - ¼x)⁴ is 16 - 8x + 3x² - x³/8 + x⁴/256.