Problem 26 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x-2)^{5} $$

Step-by-Step Solution

Answer

The expanded form of \( (x-2)^5 \) is \( x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32 \)

Step 1: Write out the Binomial Theorem

The general form of the binomial theorem is : \[ (a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + b^n \]. For this problem we have \(a = x\), \(b = -2\), and \(n = 5\).

Step 2: Apply the theorem

Substitute \(a = x\), \(b = -2\), and \(n = 5\) into the binomial theorem: \[ (x - 2)^5 = x^5 + {5\choose1}x^{5-1}(-2) + {5\choose2}x^{5-2}(-2)^2 + {5\choose3}x^{5-3}(-2)^3 + {5\choose4}x^{5-4}(-2)^4 + (-2)^5 \]. Remembering there can be negativity due to negative sign.

Step 3: Substitute and simplify terms

Substitute based on the binomial coefficient: \[ (x - 2)^5 = x^5 - 5x^4*2 + 10x^3*4 - 10x^2*8 + 5x*16 - 32 \]. Simplify the equation by multiplying: \[ (x - 2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32 \]