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Wize High School Grade 12 Pre-Calculus Textbook > Permutations and Combinations [BC/AB]

Binomial Theorem

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Binomial Theorem

Starting with a triangle made of 1's, we can see a very interesting pattern emerging by adding adjacent numbers.

11    11    2    11    3    3    11    4    6    4    1\begin{array}{c} 1 \\ 1 \ \ \ \ 1 \\ 1 \ \ \ \ 2 \ \ \ \ 1 \\ 1 \ \ \ \ 3 \ \ \ \ 3 \ \ \ \ 1 \\ 1 \ \ \ \ 4 \ \ \ \ 6 \ \ \ \ 4 \ \ \ \ 1 \end{array}
We can indefinitely continue this pattern by adding adjacent numbers to form the next number in the row. This pattern is known as Pascal's Triangle and tells us the binomial coefficients.

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Binomial Expansion

The Binomial Theorem lets us expand binomials easily by using the correct row of Pascal's Triangle.

(x+y)n=∑k=0n(nk)xn−kyk\boxed{(x+y)^n=\sum_{k=0}^n \binom nkx^{n-k}y^k}

Example

Expand:(x+2)3(x+2)^3

Using the 4th row of Pascal's Triangle.


11    11    2    11    3    3    11    4    6    4    1\begin{array}{c} 1 \\ 1 \ \ \ \ 1 \\ 1 \ \ \ \ 2 \ \ \ \ 1 \\ \bold{1 \ \ \ \ 3 \ \ \ \ 3 \ \ \ \ 1 } \\ 1 \ \ \ \ 4 \ \ \ \ 6 \ \ \ \ 4 \ \ \ \ 1 \end{array}


(x+2)3=1×x3×20+3×x2×21+3×x1×22+1×x0×23=x3+6x2+12x+8(x+2)^3= \bold1 \times x^3\times 2^0 + \bold3 \times x^2 \times 2^1+\bold3 \times x^1 \times 2^2 + \bold1 \times x^0 \times 2^3 = x^3+6x^2 +12x +8

Example: Pascals Triangle

What is the the 9th row of Pascal's Triangle?


11    11    2    11    3    3    11    4    6    4    11    5    10    10    5    11    6    15    20    15    6    11    7    21    35    35    21    7    11    8    28    56    70    56    28    8    1\begin{array}{c} 1 \\ 1 \ \ \ \ 1 \\ 1 \ \ \ \ 2 \ \ \ \ 1 \\ 1 \ \ \ \ 3 \ \ \ \ 3 \ \ \ \ 1 \\ 1 \ \ \ \ 4 \ \ \ \ 6 \ \ \ \ 4 \ \ \ \ 1 \\ 1 \ \ \ \ 5 \ \ \ \ 10 \ \ \ \ 10 \ \ \ \ 5 \ \ \ \ 1 \\ 1 \ \ \ \ 6 \ \ \ \ 15 \ \ \ \ 20 \ \ \ \ 15 \ \ \ \ 6 \ \ \ \ 1 \\ 1 \ \ \ \ 7 \ \ \ \ 21 \ \ \ \ 35 \ \ \ \ 35 \ \ \ \ 21 \ \ \ \ 7 \ \ \ \ 1 \\ \bold{1 \ \ \ \ 8 \ \ \ \ 28 \ \ \ \ 56 \ \ \ \ 70 \ \ \ \ 56 \ \ \ \ 28 \ \ \ \ 8 \ \ \ \ 1} \end{array}

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Practice: Binomial Expansion

Expand (w−z)4(w-z)^4 using Pascal's Triangle.