01Abinomialthm

= The Binomial Theorem =

Binomial expansions are the result of a bracket containing two terms being raised to a power. (bi = 2, nomial means term or name)



Arranging the __coefficients__ of these expansions into rows, forms ** Pascal's Triangle **. Each value is the __sum__ of the __two__ numbers above it in the previous row.

The binomial expansion of (x + y) 5 can be obtained from the next row of Pascal's Triangle:

Notes:
 * The indices in each term add to 5
 * The power of x starts at 5 and decreases to 0,
 * The power of y starts at 0 and increases to 5
 * The number of terms is 6 (numbered from 0 to 5)

Use Pascal's Triangle to find the expansion of (2x – 3y) 4.
 * Example **


 * Solution:**

Combinations

Recall ** Combinations ** (from Probability)

math \left( \begin{matrix} n \\ r \\ \end{matrix} \right) = \,^nC_r = \dfrac{n!}{\big( n-r \big) ! \; r!} math

For example:

math ^5C_3 = \dfrac{5!} { 2! \, 3!} = 10 math

Your calculator can find combinations using the **nCr** function. Go to the **MTH** tab of the virtual keyboard and then choose the **CALC** option.

For 5 C 3, type: **nCr**(5, 3)

You should get the result of 10.


 * Pascal's Triangle ** can also be obtained using combinations, where n is the row and r is the entry in that row (starting with 0)

Notes:
 * The indices in each term add to n
 * The power of x starts at n and decreases to 0, while the power of y starts at 0 and increases to n
 * The number of terms is n + 1 (numbered from 0 to n)

You can confirm this result on your calculator using the **expand** function from the **ACTION** menu, **TRANSFORMATION** submenu.

Type in: **expand(** (2x – 3)^4 **)**

Finding a Particular Term To find the second term, use r = 1 To find the third term, use r = 2 etc

Example 3 (Eg 5 p4)

Evaluate the term that is independent of x in the expansion of: math \Big( x^3 + \dfrac{1}{x^2} \Big)^5 math

"independent of x" means the constant term that has no x in it {ie the term with x 0 }

The powers of x for each term will be:

Your Calculator can expand binomials

From the ACTION menu, TRANSFORMATION submenu, select EXPAND

Then enter the expression to be expanded.

**expand(** (2x – 3) 4 ** ) **

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