Binomial Expansion

When the exponent is a positive integer:
( + )

When the exponent is negative or rational:
/
( 1 + )

( 1 "); if (term.charAt(0) == '-') document.write(" - " + term.substr(1) + " )"); else document.write(" + " + term + " )"); if (d==1) { document.write("" + n + "
"); document.write(""); document.write(""); document.write(""); document.write("
=    1    +    (" + n + ")(" + term + ")    +    (" + n + ")(" + (n-1) + ")(" + term + ")2
2!
"); document.write(""); document.write(""); document.write(""); document.write(""); document.write(""); document.write("
       +    (" + n + ")(" + (n-1) + ")(" + (n-2) + ")(" + term + ")3    +    (" + n + ")(" + (n-1) + ")(" + (n-2) + ")(" + (n-3) + ")(" + term + ")4    +    ...
3!4!
"); } else { document.write("" + n +"/" + d + "
"); document.write(""); document.write(""); document.write(""); document.write("
=    1    +    (" + n + "/" + d + ")(" + term + ")    +    (" + n + "/" + d + ")(" + (n-d) + "/" + d + ")(" + term + ")2
2!
"); document.write(""); document.write(""); document.write(""); document.write(""); document.write(""); document.write(""); document.write("
       +    (" + n + "/" + d + ")(" + (n-d) + "/" + d + ")(" + (n-2*d) + "/" + d + ")(" + term + ")3
3!
       +    (" + n + "/" + d + ")(" + (n-d) + "/" + d + ")(" + (n-2*d) + "/" + d + ")(" + (n-3*d) + "/" + d + ")(" + term + ")4    +    ...
4!
"); } document.write("
provided |" + term + "| < 1"); document.write("
"); } //end of function expandrational function factorial(n) { if (n > 0) return n*factorial(n - 1); else return 1; } //end of function factorial function nCr(n, r) { return factorial(n)/factorial(n - r)/factorial(r); } //end of function nCr //-->
Exercises for students
Solve the following problems on paper and check your expansions using the above application:

Expand (2x - 3y)4
Expand (2x - 3y)5
Expand (2x - 3y)6
Expand (1 + 2x)-1 as a series in ascending powers of x up to the term in x4
Expand (1 - 2x)-1 as a series in ascending powers of x up to the term in x4
Expand (1 - 2x)-2 as a series in ascending powers of x up to the term in x4
Expand (1 - 2x)-3 as a series in ascending powers of x up to the term in x4
Expand (1 - 3x)1/3 as a series in ascending powers of x up to the term in x3
Expand (1 - 3x)-1/3 as a series in ascending powers of x up to the term in x3
Expand (1 - 3x)1/2 as a series in ascending powers of x up to the term in x3
Expand (1 - 3x)2/3 as a series in ascending powers of x up to the term in x3
Expand (1 - 2x^2)-3 as a series in ascending powers of x up to the term in x6
Expand (1 + 1/x)-3 as a series in ascending powers of 1/x up to the term in 1/x3
Expand (1 + 1/x^2)-3 as a series in ascending powers of 1/x up to the term in 1/x6

Find the coefficient of t3 in the expansion of (2 - 5t)8 .

Express (1 - 3x)-1/2 as a series of ascending powers of x up to and including the term in x4 .



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