Expanding (2x - 3)⁵

Discrete Math · Axiom Academy

Apply the binomial theorem step-by-step to expand a polynomial with a negative term. Excellent work! You've successfully expanded (2x - 3)⁵ using the binomial theorem. Here's what we learned: Binomial Theorem Formula: (a + b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ, where C(n,k) are binomial coefficients. Handling Negative Terms: When b is negative (like -3), remember that (-3)ᵏ alternates signs: positive for even k, negative for odd k. Binomial Coefficients: C(5,0)=1, C(5,1)=5, C(5,2)=10, C(5,3)=10, C(5,4)=5, C(5,5)=1. Notice the symmetry! Power Rules: (2x)⁵⁻ᵏ means the power of x decreases from 5 to 0, while the power of -3 increases from 0 to 5. Sign Pattern: With (2x - 3)⁵, signs alternate: +, -, +, -, +, - because the exponent on -3 goes from 0 to 5. The binomial theorem is a powerful tool for expanding expressions without multiplying term by term. Master the pattern of coefficients and carefully track negative signs!

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