St Columba s High School Advanced Higher Maths The Binomial Theorem

Transcription

1 St Columba s High School Advanced Highe Maths The Binomial Theoem Contents 1 Intoduction to Binomials Factoial Notation Combinations and and n C 4 Pascal s Tiangle & Binomial Coefficient 6 The Binomial Theoem 9 6 Geneal Tem of the Binomial Expansion 1 7 Appoximations 1 1

2 1 Intoduction to Binomials A binomial is an algebaic expession fo the sum o diffeence of stictly two tems. Fom Highe and National maths, the tem tinomial will be familia to descibe an expession that is the sum o diffeence of thee tems, fo example x + x +. The following expessions ae examples of binomials: x + y, x 7y, 6 8y Peviously, to expand a binomial of the fom x + y the backets would be witten out fully as and then multiplied out as follows: x + yx + yx + y x + yx + yx + y x + yx + xy + y x + x y + xy + x y + xy + y x + x y + xy + y. When consideing binomials aised to highe powes, eg x + y 9, witing out the backets in full and multiplying out as shown above becomes vey time consuming. The binomial theoem povides an efficient way of witing out the expansion of these backets. Howeve, some definitions ae necessay pio to consideing the binomial theoem and binomial expansions.

3 Factoial Notation Factoials Factoials ae a quick way of witing down the poduct of all of the integes stating with the given numbe. Definition 1. The poduct of the integes n! nn 1n... 1 whee n! is ponounced n factoial. Definition. n! n n 1! Definition. 0! 1 Example 1. Find 4!. Example. Find 7!. 4! ! Combinations and and n C Anothe notation that is used in the binomial theoem is n C. This allows us to wok out the numbe of possible ways of choosing components fom a total of n. Fo example, conside the integes 1 to. How many diffeent combinations of numbes can be made fom these?

4 Hence, thee ae 10 possible ways of combining thee numbes out of. Howeve, thee is a moe efficient way to calculate these combinations without witing out a list in full. Definition 4. n C n! n!! An altenative notation fo this is: n n! n!! Fom the discussion above, the numbe of possible ways of combining thee numbes out of is!!! 10 Example. Find the value of 7 C 4 Example 4. Find 7 C ! 4!7 4!

5 10 10! 10!!

6 4 Pascal s Tiangle & Binomial Coefficient Anothe means of finding coeffiecients in the binomial expansion is to use Pascal s tiangle. This is fomed by adding the two tems diectly above to make the next ow. The tiangle stats at ow 0. 1 Row Row Row 1 1 Row Row Row This tiangle can be linked to the n notation as shown: Row 0 Row 1 Row Row Hence, means the second enty in Row of Pascal s tiangle. Note that the initial enty in each ow is classed as the 0th enty. Looking at the taingle, it is clea that this is. This can be seen to 6

7 be tue as!!! Definition. As each tem in Pascal s tiangle is obtained by adding togethe the two enties diectly above, in geneal: n n n n Example. Solve fo n N whee N is the set of natual numbes which includes only all positive integes. n n!!n! nn 1n 1 1n n 1 nn 1 n n 6 n n 6 0 n n + 0 n o n Howeve, as n N, the solution must be positive and so n. 7

8 n Example 6. Fo n N, solve 1 n + 8 n n n n + 1! 8!n + 1! n + 1nn 1n 1 8 1n 1n 1 n + 1n 8 n + n 6 n + n 6 0 n 7n n 7 o n 8 Howeve, as n N, the solution must be positive and so n 7. 8

9 The Binomial Theoem A binomial of the fom x + y n, can be expanded multiplied out using the coefficients fom Pascal s tiangle. Fo example x + y 4 can be expanded using the numbes found in the fouth ow of Pascal s tiangle as shown: x + y 4 1x 4 + 4x y + 6x y + 4xy + 1y 4 In geneal, fo any binomial, the expansion can be found using the binomial theoem. Definition 6. The Binomial Theoem n x + y n n x n y 0 n x 0 n n + x 1 n 1 n y + x n y n + + y n n whee is ponounced sigma and is the notation to define the sum of tems this notation was used befoe in calculating standad deviation. Note that the coefficients in the expansion come fom the ow of Pascal s tiangle that coesponds to the powe of the binomial. As the powes of the fist tem incease, the powes of the second tem decease. The sum of the total powes in each pat of the expansion must equal the powe of the oiginal binomial. 9

10 Example 7. Expand x + y 4 using the Binomial Theoem. x + y x x 4 y 4 x 1 4 y + x y 4 + xy 4 + y 4 4 x x y + 6 4x y + 4 8xy + 16y 4 x 4 + 8x y + 4x y + xy + 16y 4 Example 8. Expand x y using the binomial expansion. x y x y 0 x 0 + x 1 y + x y + y 1 x + x y + x y + y 7x 4x y + 6xy 8y 10

11 Example 9. Find the coefficient of x 6 in the expansion of x 9. x The coefficient of x 6 is calculated as follows: x 6 9!!6! x !!6! x6 84 7x 6 9 y 7x 6 48x 6 Hence the coefficient of x 6 in the expansion of x 9 is

12 Example 10. Expand y y. Using the Binomial theoem alongside the enties in the thid ow of Pascal s tiangle gives: 0 y y 1 y + y y + y y + 1 y y 6 y y + 1 y y 8 y y 6y + 1 y 8 y 1

13 6 Geneal Tem of the Binomial Expansion Definition 7. The geneal tem of x + y n is given by : n x n y Example 11. Find the coefficient of x y 4 in the expansion of x + y 7 The geneal tem is: x + y 7 7 x 7 y As the coefficient of x y 4 is to be found and the geneal tem has powes x 7 y, by equating powes, it is clea that 4. Theefoe, the coefficient is: 7 7! 4!7 4! Example 1. Find the coefficient of x 4 in the expansion of 1 + x. The geneal tem is 1 + x 1 x x 1

14 Hence, fo the coefficient of x 4, must be equal to 4 and so. Hence, the coefficient of x 4 is 1. x 4!!1! x 4 1x 4 Example 1. Find the coefficient of y in the expansion of y 1 y. The geneal tem is y 1 y y 1 y y 1 y 1 y Hence, fo the coefficient of y, and so 0. 1 Hence, the coefficient of y is y Note that in cases whee the tem independent of x is to be found, it is necessay to look at the geneal tem and identify the value of that would make the powe of x equal zeo. 14

15 7 Appoximations The binomial theoem can also be used with decimal numbes to find appoximations. Example 14. Find 1.0 coect to thee decimal places to decimal places Example 1. Find 0.9 coect to two decimal places to decimal places 1