Practice Flashcards
Flip to reveal answersIn (3x − 2)⁵ a student writes the x² term as ⁵C₃ x² · 2³. What is wrong?
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All 10 Flashcards — Finding a term
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Question
In (3x − 2)⁵ a student writes the x² term as ⁵C₃ x² · 2³. What is wrong?
Answer
Two errors — the sign and the 3. Here a = 3x (not x) and b = −2 (not 2). Correct: ⁵C₃ (3x)²(−2)³ = 10 × 9 × (−8) x² = −720x². Always raise the WHOLE bracket term — number, variable and sign — to its power. (General term: ⁿCᵣaⁿ⁻ʳbʳ.)
Question
How do you find a specific term without expanding?
Answer
Use the general term ⁿCᵣ aⁿ⁻ʳ bʳ; set the exponent of x equal to the power you want, solve for r, then compute that one term.
Question
How do you find one coefficient?
Answer
Write the general term, find the r that gives that power of x, and compute the coefficient (raising the whole coefficient/sign to the power).
Question
What does 'term independent of x' (constant term) mean?
Answer
The power of x is 0. Set the exponent of x to 0, solve for r, then compute that term.
Question
Given a coefficient, how do you find an unknown constant?
Answer
Write that coefficient via the general term, set it equal to the given value, and solve. E.g. (x+k)⁷ coeff x⁵ = 63 → 21k² = 63 → k = ±√3.
Question
Why do you sometimes get ± for the unknown?
Answer
An even power of the unknown (e.g. k²) gives two values. Check for a restriction like 'k > 0' before keeping both.
Question
How do you find an unknown power n?
Answer
Use the simplest coefficient: ⁿC₂ = n(n − 1)/2 gives a quadratic in n; solve for the positive integer.
Question
From the first terms of (1 + kx)ⁿ, how do you find n and k?
Answer
Use ⁿC₁k = (x coefficient) and ⁿC₂k² = (x² coefficient); eliminate k and solve for n, then k.
Question
Find the coefficient of x⁴ in (2x − 3)⁶.
Answer
r = 2: ⁶C₂(2x)⁴(−3)² = 15 × 16 × 9 = 2160.
Question
Two unknowns and two coefficient conditions — fastest method?
Answer
Form both equations and divide one by the other to eliminate a variable.
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Full study notes for Finding a term
Topic 1.9 hub
Binomial theorem
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Math AA SL exam skills
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