Properties of coefficients

1. Properties of coefficients • Expand the following = 1 = = = = Back to year 7 Now in year 12 Pascal’s triangle was Pascal’s triangle is 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 Sum = 1 Sum = 2 Sum = 4 Sum = 8 Sum = 16 Sum = 32 Sum = 64 = = = = = = =

2. Expansion of Notice the “term number” is always one more than the “other numbers”

3. Expansion of Here, OR Note, smaller font Now for some more Greek, back in yr8 we met  in the stats topic. Here (and there)  means sum or total.

4. Example 1 Expand the binomial product (1 + x)6. Find the sum of its coefficients. c) Write the binomial expansion in sigma notation. a) b) Sum of coefficients of (1 + x)nis as n = 6, the sum is 26 or 2n 64 c) Using

5. Example 2 How many terms in the expansion of (1 + x)15. Evaluate the coefficient of x12. c) Find the 8th term in the expansion of (1 + x)15. a) There are n + 1 terms in the expansion of (1 + x)n. there are 16 terms in (1 + x)15. . b) The general term of a binomial expansion is The expansion of (1 + x)15the term involving x12is 15C12x12. The coefficient is 15C12= 455. c) The 8th term means k = 7 =

6. Today’s work Exercise 10.2 Page 477 → 478 Q1c Q2, 3, 5, 6a + b Q4, 8a + c Q7, 9a + e Q10 Yesterday’s work Exercise 10.1 Page 472 → 473 Q1→3 a, c, e . . . Q5→9 Q10→13 a, c + e Q14 + 15