Use the binomial theorem to expand (2-1/4X)⁴
I assume you mean
(2 - x/4)^4 not ( 2 - 1/(4x) )^4
= 16 - (4 * 8 * - x/4) + 6*4*x^2/16 - 4*2*x^3/64 + x^4/256
= 16 - 8x + (3/2)x^2 - (1/8)x^3 + (1/256)x^4
I mean (2 - 1/4X)⁴
To expand the expression (2 - 1/4X)⁴ using the binomial theorem, you can follow these steps:
Step 1: Identify the values of a and b in the binomial expression.
In this case, a = 2 and b = -1/4X.
Step 2: Identify the power or exponent of the binomial expression.
The power/exponent in this case is 4.
Step 3: Apply the binomial theorem formula.
The formula for the binomial theorem is:
(x + y)ⁿ = nC0 * xⁿ * y⁰ + nC1 * xⁿ⁻¹ * y¹ + nC2 * xⁿ⁻² * y² + ... + nCr * xⁿ⁻ʳ * yʳ + ... + nCn * x⁰ * yⁿ
Where:
- n is the power/exponent of the binomial expression.
- x is the first term (a).
- y is the second term (b).
- nCk is the binomial coefficient, which represents the number of ways to choose k items out of a set of n items. It is calculated as n! / (k! * (n-k)!), where ! denotes factorial.
Step 4: Put the values into the binomial theorem formula.
Using the formula, and the values we identified in steps 1 and 2, we can expand (2 - 1/4X)⁴ as follows:
(2 - 1/4X)⁴ = 4C0 * 2⁴ * (-1/4X)⁰ + 4C1 * 2³ * (-1/4X)¹ + 4C2 * 2² * (-1/4X)² + 4C3 * 2¹ * (-1/4X)³ + 4C4 * 2⁰ * (-1/4X)⁴
Step 5: Simplify and Compute.
Now, we simplify each term according to the binomial coefficients:
= 1 * 2⁴ + 4 * 2³ * (-1/4X) + 6 * 2² * (1/16X²) + 4 * 2¹ * (-1/64X³) + 1 * 2⁰ * (1/256X⁴)
Finally, we simplify further:
= 16 + 8 * (-1/4X) + 3/4 * (1/4X²) + 1/4 * (-1/8X³) + 1/256X⁴
Thus, the expanded form of (2 - 1/4X)⁴ is 16 - 2/X + 3/16X² - 1/32X³ + 1/256X⁴.