how do i use the binomial theorem?

the review sheet i have says to expand this:

(x + 2y)^4

how would i do that? it's so confusing.

also, another question asks to find the term containing y^3 in this: (sqroot2 + y)^12, how do i do that?

1 Answer

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  • 1 decade ago
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    General solution to (a+b)ⁿ

    Note: nCr (n choose r) = n! / (r!(n-r)!)

    (a+b)ⁿ = nC0 aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + ... + nC(n-1) abⁿ⁻¹ + nCn bⁿ

    (a+b)ⁿ = Σ(i=0..n) nCi aⁿ⁻ⁱ bⁱ

    (x + 2y)⁴ = 4C0 x⁴ + 4C1 x³(2y) + 4C2 x²(2y)² + 4C3 x(2y)³ + 4C4 (2y)⁴

    (x + 2y)⁴ = 1 x⁴ + 4 x³(2y) + 6 x²(4y²) + 4 x(8y³) + 1 (16y⁴)

    (x + 2y)⁴ = x⁴ + 8 x³y + 24 x²y² + 32 xy³ + 16y⁴

    ------------------------- -------------------------

    (√2 + y)¹² = Σ(i=0..12) 12Ci √2¹²⁻ⁱ yⁱ

    The term containing y³ occurs when i = 3

    12C3 (√2)¹²⁻³ y³ = 220 (√2)⁹ y³ = 220 √(2⁹) y³ = 220*2⁴√2 y³ = 3520√2 y³

    So, term containing y³ is : 3520√2 y³

    .

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