Find the term in the expansion of (2x-5)^6 which have 1.greatest binomial coefficient, 2.greatest numerical coefficient, 3. Algebraically greatest coefficient & 4. Algebraically least coefficient.
Answers
Answer:
The expansion of:
(2x - 5)⁶ = ⁶C₀ (2x)⁶ + ⁶C₁ (2x)⁵(-5) + ⁶C₁ (2x)⁵(-5) + ⁶C₂ (2x)⁴(-5)² + ⁶C₃ (2x)³(-5)³ + ⁶C₄ (2x)²(-5)⁴ + ⁶C₅ (2x)(-5)⁵ + ⁶C₆ (-5)⁶
1. greatest binomial coefficient: ⁶C₀ (2x)⁶
2. greatest numerical coefficient: ⁶C₃ (2x)³(-5)³ and ⁶C₄ (2x)²(-5)⁴
3. Algebraically greatest coefficient: ⁶C₄ (2x)²(-5)⁴
4. Algebraically least coefficient: ⁶C₀ (2x)⁶
Answer:
Step-by-step explanation:
Our aim is to find the term in the expansion of (2x-5)^6 which have the given properties.
Our first line asks for a greatest binomial coefficient. Hence, since we have to find the term in expansion of (2x-5)^6 we have the following expansion of the term:
(2x - 5)⁶ = ⁶C₀ (2x)⁶ + ⁶C₁ (2x)⁵(-5) + ⁶C₁ (2x)⁵(-5) + ⁶C₂ (2x)⁴(-5)² + ⁶C₃ (2x)³(-5)³ + ⁶C₄ (2x)²(-5)⁴ + ⁶C₅ (2x)(-5)⁵ + ⁶C₆ (-5)⁶
Now, in order to compute all lines we get that for the first question, the greatest binomial coefficient is ⁶C₀ (2x)⁶.
In the second line, the greatest numerical coefficient are ⁶C₃ (2x)³(-5)³ and ⁶C₄ (2x)²(-5)⁴.
Third question, our algebraically greatest coefficient is ⁶C₄ (2x)²(-5)⁴.
At last but not least, the algebraically least coefficient is equal to ⁶C₀ (2x)⁶.
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