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Lesson 3- Polynomials

Lesson 3- Polynomials

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Lesson 3- Polynomials

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

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

  3. Multiplying Polynomials • Expand all the terms ML3 MH

  4. Dividing Polynomials • This is trickier than multiplication • There are two main ways • Long Division • By Inspection ML3 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” ML3 MH

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

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

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

  11. Algebraic long division Multiply –x by x + 2 and subtract ML3 MH

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

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

  14. Algebraic long division Multiply x + 2 by 1 and subtract ML3 MH

  15. Algebraic long division The quotient is 2x² - x + 1 The remainder is –1. ML3 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 ML3 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 quotient by ax² + bx + c Let the remainder be d. 2x³ + 3x² - x + 1 = (x + 2)(ax² + bx + c) + d ML3 MH

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

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

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

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

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

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

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

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

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

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