Expression Term Equation Coefficient Identity Function Polynomial Root Lesson Objective Understand the meaning of the words: Be able to simplify polynomials, extend to adding, subtracting, multiplying and dividing them Expression Term Equation Coefficient Identity Function Polynomial Root Be able to manipulate polynomials

Expression Term Equation Coefficient Identity Function Polynomial Root An algebraic statement made up of variables and numbers with their respective powers along with a some mathematical operations of any type An statement consisting of variables and numbers with their respective powers that are only multiplied or divided An expression that contains an equals sign The number part of a term A statement of equivalence between two expressions that is always true for the same inputted value Another name for a formula An expression that can be simplified so that it contains only terms in one variable with positive whole number powers The name given to a solution of an equation

Expressions, Terms, Coefficients and Polynomials An expression is a mathematical ‘sentence’. It contains terms that are separated by + and – signs. A term is a ‘word’ in the ‘sentence’. Eg 3x2 – 4y + 2x + 3xy – 4x2y + 2y + Expressions that are of the form: a0xn + a1xn-1+ a2xn-2 + a3xn-3 + ………….. + an-1x + anx0 are called polynomials

3x2 + 2xy – 3y + - 2xy2 + - 0.7 How many terms? What is the coefficient in x2? What is the coefficient in y? What is the coefficient in ? What is the constant term? Which of the following expressions are polynomials? 2x3 + x2 + x b) 3x + 1 c) 6 3x + x-5 e) sin(x) + 2 f) x3 g) x7 + 1/2x5 - 3x h) i) π

Lesson Objectives: Simplify: Be able to simplify polynomial expressions Be able to add, subtract and multiply polynomial expressions Simplify: 3x + 4x2 – 2y + 3y -4x + 5x2 + 2xy – 3yx + 5x2y

Expressions are simplified by collecting together ‘like terms’ ‘Like terms’ are those that contain exactly the same letters and powers Eg Simplify 3x + 2xy + 4x + 3yx + 2x2 Simplify 5xy2 + 3xy + 2yx + 2x2y + 3xy2 Simplify 3x + 2y – 4x + 3 – 5x2 – 3x2 + 4y + 7

Simplify: 3x + 4y - 2x + 6y 3 – 4x + 6 + 2x 2x + 3x2 – 3x – x2 2y + 3y – 4x – 5y + x 3xy + 2yx + 3x – 5y – 6x – 2yx 7x2 + 3x + 4x – 5x2 + 2x 2x2 – 3x + 2x2y + 3xy – 4yx2 – 2x2 – 5yx 3x + 2x2 – 4x + 3x2 – 2x + x -5xy + 3yx + 2xy – xy + x + 3x 2x2 + 3x – 4x2 – 3x + x2 – 2x + 3x2

Multiplying Polynomials A= 2x + 3 C = 3x - 5 B = 3x2 – 2x – 5 D = 2x2 + x – 4 Find: A + C b) A + B c) A – B d) AC e) A2 f) C2 D - B h) 2A - C i) BC j) BD k) A + BD l) B2 m) A3 n) AD – BC o) A ÷ D

Multiplying Polynomials A= 2x + 3 C = 3x - 5 B = 3x2 – 2x – 5 D = 2x2 + x – 4 Find: A + C b) A + B c) A – B d) AC e) A2 f) C2 D - B h) 2A - C i) BC j) BD k) A + BD l) B2 m) A3 n) AD – BC o) A ÷ D 5x - 2 3x2 - 2 -3x2 + 4x + 8 6x2 – 9x - 15 4x2 + 12x + 9 9x2 – 30x + 25 -x2 + 3x + 1 X + 11 9x3- 21x2- 5x+ 25 15x4- x3- 24x2 + 3x +20 6x4- x3- 24x2 + 5x +23 9x4- 12x3- 26x2 + 20x + 25 -5x3 + 29x2 - 37 8x3 + 36x2 + 54x + 27

Lesson Objectives: Dividing polynomials Common misconceptions:

Dividing Polynomials Find 65325 ÷ 4 What about ( ) ( ) ÷ ? .

Find the missing factor if: Divide by Divide by .

+ - × ÷ x2+5x - 6 x + 3 2x3- x2 x - 1 3x2+8x+4 x2 + x x2 + 4x + 3 We need to be able to accurately add, subtract, multiply and divide expressions: x2+5x - 6 x + 3 2x3- x2 x - 1 3x2+8x+4 x2 + x x2 + 4x + 3 x3+ 2x2- 4x + 1 x3+ 3x2+ 3x + 1 x + 1 + - × ÷

Lesson Objective Factorising single and double brackets

Find 1) (x + 2)(x + 5) 2) (2x + 1)(x + 4) 3) (x2 – 3)(x2 + 2x + 4) 4) (x + y)(2x – 3y + 4) 5) (2x4 – 2x2 + 3x – 5) ÷ (2x + 1)

4x – 6 2) 9x + 12 3) 8x – 12 4) 12x – 9 4x2 + 2x 6) 9x2 – 6x 7) 12x2 + 15x 8) 12x + 8x2 20x – 15x2 10) 8xy + 12x2y 11) 8x2y – 6xy2 12) 9x2 – 27x + 12 13) x2 + 7x + 12 14) x2 + 9x + 20 15) x2 – 7x – 30 16) x2 + 3x – 18 x2 – 9 18) x2 + 7x – 18 19) x2 + x – 12 20) x2 - 64

21) 2x2 + 5x + 3 22) 3x2 + 5x + 2 23) 2x2 + 9x + 10 24) 5x2 + 6x + 1 25) 2x2 + 7x + 3 26) 3x2 + 9x + 20 27) 3x2 + 10x + 7 28) 7x2 + 23x + 6 29) 4x2 + 4x + 1 30) 4x2 + 5x + 1 31) 4x2 – 25 32) 16x2 - 100

Lesson Objective the Remainder Theorem

3x + 4 = 2x – 6 2(3x + 1) = 6 + 2(x – 1) ½(x + 6) = x + 1/3(2x – 5) Page 10 and 11 Exercise Book