Created by Mr. Lafferty Maths Dept. Binomial Expansion Binomial Expansion Complex Numbers www.mathsrevision.com Fundamental Theorem of Algebra 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Learning Intention Success Criteria 1. We are learning how Pascal’s triangle can help us expand polynomials. 1. Be able to construct Pascal’s triangles. www.mathsrevision.com 2. Be able to expand polynomials brackets using Pascal’s triangle. 5-Sep-18 Created by Mr. Lafferty Maths Dept.

(x + y)n Binomial Expansion The Binomial expansion provides a method of expanding (x + y)n in powers of x where the index n is a positive integer. 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Can you find a connection ? Find the next two rows. Binomial Expansion Expand using National 5 techniques (1 + x)0 (b) (1 + x)1 (c) (1 + x)2 (d) (1 + x)3 (e) (1 + x)4 1 5 10 10 5 1 1 6 15 20 15 6 1 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Binomial Expansion The sum of powers of x and y sum to 3 (n) Coefficients are from Pascal’s triangle Binomial Expansion Using Pascal’s triangle we can calculate more complicated expressions. Example : Expand (x + y)3 As powers of x decrease powers of y increase. 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Exercise 1 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion The term factorial is a short hand way of writing and is written as r! 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Calculator Button ! Three ways of representing the Binomial coefficient of a term The word binomial means “2 terms “ and the binomial expansion for (1 + x)n is written as 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Also 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Expand (1 - x)4 Calculator Button ! OR 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Example : Expand (4 + x)5 Calculator Button ! 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Example : Find the value of (1.9)4 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Binomial Expansion Exercise 2 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers We know how to solve quadratic equations by factorising or quadratic formula. When the roots are real b2 -4 ac ≥ 0 When b2 -4 ac < 0 then roots are complex The concept of an Imaginary Number helps us to find the complex roots of an quadratic equation 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers Suppose there exists a non-zero number i such Then we can find complex roots for b2 -4 ac < 0 Example : 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers A number of the form where b is real is called a IMAGINARY NUMBER A number of the form where x,y are real is called a COMPLEX NUMBER 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers A complex number is normally represented by the letter z Real part z : Imaginary part z : WE CAN NOW SOLVE ANY QUADRATIC 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Example : Solve Note If one root of a quadratic is then the other will be These are called the Complex Conjugates of each other. 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Points to Note Sum of complex roots are REAL and equal to the NEGATIVE of the coefficient of x Product of complex roots are REAL and equal to the constant c in quadratic equation Complex roots are Complex Conjugates of each other If a root is Conjugate is 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Example : Find Solve 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Exercise 1 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Basic Rules Addition and Subtraction 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Basic Rules Conjugate property Product and Division 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers Using Binomial Expansion Powers Basic Rules 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Basic Rules Root x is real 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Exercise 2 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Geometrical Representation Im (y) Argand Diagram Re (x) 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Addition Im (y) Re (x) If and then 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Subtraction If and then 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Using Basic Trig MODULUS If 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Modulus & Argument Find the modulus and argument 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Modulus & Argument Find the modulus and argument 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Complex Numbers Modulus & Argument Useful for x and ÷ Exercise 3 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers Fundamental Theorem of Algebra All complex polynomial equations have a least one complex root. This is called the Fundamental Theorem of Algebra Consequently, if a polynomial is of degree n, then there are precisely n roots in the set of complex numbers. Some or all may be real, some or all maybe complex. 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers If is a root of a polynomial equation then Fundamental Theorem of Algebra If is a root of a polynomial equation then is also a root. and are therefore factors of the polynomial. More precisely is a quadratic factor. 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers Higher Synthetic ÷ Find the roots of f(-3) = 0 there Fundamental Theorem of Algebra Find the roots of f(-3) = 0 there The roots are using quadratic formula 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers Verify is a factor of and find all the other roots. Fundamental Theorem of Algebra Verify is a factor of and find all the other roots. Substitute into Binomial expansions if equal to 0 then is a root. = 0 Hence factor 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers Complex Conjugate Since then is also a factor. Fundamental Theorem of Algebra Since then is also a factor. Quadratic factor is Now divide to get other factors 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Complex Numbers The roots are a = 1 , b = -1 , c = 1 Overall roots are Fundamental Theorem of Algebra The roots are a = 1 , b = -1 , c = 1 Overall roots are using quadratic formula 5-Sep-18 Created by Mr. Lafferty Maths Dept.

Exercise 6 Complex Numbers Fundamental Theorem of Algebra 5-Sep-18 Created by Mr. Lafferty Maths Dept.