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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §6.4 Divide PolyNomials. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 6.3. Review §. Any QUESTIONS About §6.3 → Complex Rational Expressions Any QUESTIONS About HomeWork §6.3 → HW-25. §6.4 Polynomial Division.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §6.4 DividePolyNomials Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 6.3 Review § • Any QUESTIONS About • §6.3 → Complex Rational Expressions • Any QUESTIONS About HomeWork • §6.3 → HW-25

  3. §6.4 Polynomial Division • Dividing by a Monomial • Dividing by a BiNomial • Long Division

  4. Dividing by a Monomial • To divide a polynomial by a monomial, divide each term by the monomial. • EXAMPLE – Divide: x5 + 24x4 − 12x3 by 6x • Solution

  5. Example  Monomial Division • Divide: • Solution:

  6. Dividing by a Binomial • For divisors with more than one term, we use long division, much as we do in arithmetic. • Polynomials are written in descending order and any missing terms in the dividend are written in, using 0 (zero) for the coefficients.

  7. Recall Arithmetic Long Division • Recall Whole-No. Long Division Divide: Quotient Divisor Remainder Quotient 13 • Divisor 12 + Remainder 1 = Dividend = 157 • +

  8. Binomial Div.  Step by Step • Use an IDENTICAL Long Division process when dividing by BiNomials or Larger PolyNomials; e.g.; 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.

  9. Binomial Div.  Step by Step 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.

  10. Binomial Div.  Step by Step Now multiply (x+2) by 2x² and subtract

  11. Binomial Div.  Step by Step Bring down the next term, -x.

  12. Binomial Div.  Step by Step Now divide –x², the first term of –x² - x, by x, the first term of the divisor which gives –x.

  13. Binomial Div.  Step by Step Multiply (x +2) by -x and subtract

  14. Binomial Div.  Step by Step Bring down the next term, 1

  15. Binomial Div.  Step by Step Divide x, the first term of x + 1, by x, the first term of the divisor which gives 1

  16. Binomial Div.  Step by Step Multiply x + 2 by 1 and subtract

  17. Binomial Div.  Step by Step The quotient is 2x² - x + 1 The remainder is –1.

  18. Multiply (x + 3) by x, using the distributive law Subtract by changing signs and adding Example  BiNomial Division • Divide x2 + 7x + 12 by x + 3. • Solution

  19. Multiply 4 by the divisor, x + 3, using the distributive law Subtract Bring Down the +12 Example  BiNomial Division • Solution – Cont.

  20. Example  BiNomial Division • Divide 15x2− 22x + 14 by (3x− 2) • Solution • The answer is 5x −4 with R6. We can also write the answer as:

  21. Example  BiNomial Division • Divide x5− 3x4− 4x2 + 10x by (x − 3) • Solution • The Result

  22. Divide 3x2− 4x− 15 by x− 3 • SOLUTION: Place the TriNomial under the Long Division Sign and start the Reduction Process Divide 3x2 by x: 3x2/x = 3x. Multiply x – 3 by 3x. Subtract by mentally changing signs and adding −4x + 9x = 5x.

  23. Divide 3x2− 4x− 15 by x− 3 • SOLUTION: next divide the leading term of this remainder, 5x, by the leading term of the divisor, x. Divide 5x by x: 5x/x = 5. Multiply x – 3 by 5. Subtract. Our remainder is now 0. • CHECK: (x− 3)(3x + 5) = 3x2− 4x− 15 The quotient is 3x + 5.

  24. Divisor Dividend Quotient Remainder Formal Division Algorithm • If a polynomial F(x) is divided by a polynomial D(x), with D(x) ≠ 0, there are unique polynomials Q(x) and R(x) such that F(x) = D(x) • Q(x) + R(x) • Either R(x) is the zeropolynomial, or the degree of R(x) is LESS than the degree of D(x).

  25. PolyNomial Long Division • Write the terms in the dividend and the divisor in descending powers of the variable. • Insert terms with zero coefficients in the dividend for any missing powers of the variable • Divide the first terms in the dividend by the first terms in the divisor to obtain the first term in the quotient.

  26. PolyNomial Long Division • Multiply the divisor by the first term in the quotient, and subtract the product from the dividend. • Treat the remainder obtained in Step 4 as a new dividend, and repeat Steps 3 and 4. Continue this process until a remainder is obtained that is of lower degree than the divisor.

  27. Example  TriNomial Division • Divide • SOLN

  28. Example  TriNomial Division • SOLNcont.

  29. Example  TriNomial Division • Divide • The Quotient = • The Remainder = • Write the Result in Concise form:

  30. WhiteBoard Work • Problems From §6.4 Exercise Set • 30, 32, 40 • BiNomialDivision

  31. All Done for Today PolynomialDivisionin base2 From UC Berkeley Electrical-Engineering 122 Course

  32. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  33. Graph y = |x| • Make T-table

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