1. Lesson 3- Polynomials 17 May, 2015ML3 MH Objectives : - Definition - Dividing Polynomials Next Lesson - Factor Theorem - Remainder Theorem

2. Polynomial 17 May, 2015ML3 MH Real numbers called coefficients Constant n is the Degree of the polynomial

3. Multiplying Polynomials Expand all the terms 17 May, 2015ML3 MH

4. Dividing Polynomials This is trickier than multiplication There are two main ways ─ Long Division ─ By Inspection 17 May, 2015ML3 MH

5. Dividing polynomials This PowerPoint presentation demonstrates two different methods of polynomial division. Click here to see algebraic long division Click here to see dividing “in your head” 17 May, 2015ML3 MH

6. Algebraic long division Divide 2x³ + 3x² - x + 1 by x + 2 x + 2 is the divisor The quotient will be here. 2x³ + 3x² - x + 1 is the dividend 17 May, 2015ML3 MH

7. Algebraic long division First divide the first term of the dividend, 2x³, by x (the first term of the divisor). This gives 2x². This will be the first term of the quotient. 17 May, 2015ML3 MH

8. Algebraic long division Now multiply 2x² by x + 2 and subtract 17 May, 2015ML3 MH

9. Algebraic long division Bring down the next term, -x.-x. 17 May, 2015ML3 MH

10. Algebraic long division Now divide –x², the first term of –x² - x, by x, the first term of the divisor which gives –x.–x. 17 May, 2015ML3 MH

11. Algebraic long division Multiply –x by x + 2 and subtract 17 May, 2015ML3 MH

12. Algebraic long division Bring down the next term, 1 17 May, 2015ML3 MH

13. Algebraic long division Divide x, the first term of x + 1, by x, the first term of the divisor which gives 1 17 May, 2015ML3 MH

14. Algebraic long division Multiply x + 2 by 1 and subtract 17 May, 2015ML3 MH

15. Algebraic long division The remainder is –1. The quotient is 2x² - x + 1 17 May, 2015ML3 MH

16. Dividing polynomials Click here to see this example of algebraic long division again Click here to see dividing “in your head” Click here to end the presentation 17 May, 2015ML3 MH

17. Dividing in your head Divide 2x³ + 3x² - x + 1 by x + 2 When a cubic is divided by a linear expression, the quotient is a quadratic and the remainder, if any, is a constant. Let the remainder be d. Let the quotient by ax² + bx + c 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d 17 May, 2015ML3 MH

18. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d The first terms in each bracket give the term in x³x³ x multiplied by ax² gives ax³ so a must be 2.2. 17 May, 2015ML3 MH

19. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² + bx + c) + d The first terms in each bracket give the term in x³ x multiplied by ax² gives ax³ so a must be 2. 17 May, 2015ML3 MH

20. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² + bx + c) + d Now look for pairs of terms that multiply to give terms in x² x multiplied by bx gives bx² + 4x² must be 3x²3x² 2 multiplied by 2x²2x² gives 4x²4x² so b must be 17 May, 2015ML3 MH

21. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² + -1x + c) + d Now look for pairs of terms that multiply to give terms in x² x multiplied by bx gives bx² bx² + 4x² must be 3x² 2 multiplied by 2x² gives 4x² so b must be -1. 17 May, 2015ML3 MH

22. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + c) + d Now look for pairs of terms that multiply to give terms in x x multiplied by c gives cx - 2x 2x must be -x-x 2 multiplied by -x-x gives -2x so c must be 1. 17 May, 2015ML3 MH

23. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) + d Now look for pairs of terms that multiply to give terms in x x multiplied by c gives cx cx - 2x must be -x 2 multiplied by -x gives -2x so c must be 1. 17 May, 2015ML3 MH

24. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) + d Now look at the constant term 2 multiplied by 1 gives 2 2 + d must be 1 then add d so d must be 17 May, 2015ML3 MH

25. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) - 1 Now look at the constant term 2 multiplied by 1 gives 2 2 + d must be 1 then add d so d must be -1. 17 May, 2015ML3 MH

26. Dividing in your head 2x³ + 3x² - x + 1 = (x + 2)(2x² - x + 1) - 1 The quotient is 2x² - x + 1 and the remainder is –1. 17 May, 2015ML3 MH

27. Dividing polynomials Click here to see algebraic long division Click here to see this example of dividing “in your head” again Click here to end the presentation 17 May, 2015ML3 MH

28. Do the following 1. 2. 3. 4. 17 May, 2015ML3 MH Exercises C1/C2 Page 82 Ex 3A, Nos 3, 6, 9, 16 to 20