Core 1 Polynomials Dividing polynomials, Factor Theorem and Remainder Theorem. Binomial Expansion Since we’ll be talking about factorials (5! = 1×2×3×4×5 = 120) in the binomial expansion, a question to think about before we start: How many zeros are there after the last non-zero digit in 100! ?
Core 1 Polynomials Dividing polynomials, Factor Theorem and Remainder Theorem. Binomial Expansion Since we’ll be talking about factorials (5! = 1×2×3×4×5.
Dec 13, 2015
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Core 1 Polynomials
Dividing polynomials, Factor Theorem and Remainder
Theorem.
Binomial Expansion Since we’ll be talking about factorials (5! = 1×2×3×4×5 = 120) in the binomial expansion, a question to think about before we start:
How many zeros are there after the last non-zero digit in 100! ?
3 2f 3 4y x x x
2f 1 2 2 2x x x x 2f 1 4 4x x x x
Factor Theorem Remainder Theorem
Polynomial division
Polynomial multiplication
2 22 4 5 3 4x x x x x x
Polynomial division3 2Divide 3 5 14 by 2x x x x
Polynomial division3 2Divide 2 5 by 2x x x
Two to try
3 2
3 2
Divide 3 4 by 1
Divide 3 4 by 1
x x x
x x x
3 2f 3 4y x x x
2f 1 2 2 2x x x x 2f 1 4 4x x x x
Factor Theorem Remainder Theorem
Polynomial division
Factor Theorem
If (x-a) is a factor of f(x),
then f(a)=0 and x=a is a root of the equation f(x)=0.
Conversely, if f(a)=0 then (x-a) is a factor of f(x).
4 3 3 2f 5 11 21 3 2 6 7x x x x x x x x
Remainder Theorem
For a polynomial f(x),
f(a) is the remainder when f(x) is divided by (x-a).
3 2 2f 3 5 9 2 5 5 1x x x x x x x
MEI Core 1, June 10, Qn 6, 5 marks
MEI Core 1, June 09, Qn 3, 3 marks
Binomial Expansion
Magic?
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
4 4 3 2 2 3 41 4 6 4 1a b a a b a b ab b
Why Pascal’s triangle gives the binomial coefficients
4a b
The problem with relying on Pascal’s triangle
How would you find the coefficient of x7 in the expansion of 21
3 2 ?x
Pascal’s Triangle isn’t really about adding numbers – it’s
about choosing.
42C
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
53C
!
! !n
r
nC
r n r
Binomial Expansion
5a b a b a b a b a b a b
17 2Find the coefficient of in the expansi 3on of 2x x
MEI Core 1, Jan 10, Qn 8, 4 marks
MEI Core 1, June 09, Qn 5, 4 marks
MEI Core 1, Jan 08, Qn 7, 4 marks
Two much more challenging questions involving nCr
• How many anagrams of ANAGRAM do not contain adjacent As?
• How many ways are there of writing 20 as a sum of exactly 4 positive integers where order matters? (3+8+8+1 is different from 1+3+8+8)
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