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1.4 - Dividing Polynomials

1.4 - Dividing Polynomials. MCB4U. (A) Review. recall the steps involved in long division: set it up using the example of 30498 ÷ 39. (B) Division of Polynomials by Factoring. sometimes it will be easier to factor a polynomial and simply cancel common factors ex.

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1.4 - Dividing Polynomials

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  1. 1.4 - Dividing Polynomials MCB4U

  2. (A) Review • recall the steps involved in long division: • set it up using the example of 30498 ÷ 39

  3. (B) Division of Polynomials by Factoring • sometimes it will be easier to factor a polynomial and simply cancel common factors • ex.

  4. (C) Restrictions in Division • any time we divide there is always one restriction, in that you cannot divide by zero • so the denominator of a fraction or rational expression or the divisor cannot be equal to zero • so in the example above, x + 3 ≠ 0, so x ≠3. • With the example above, draw it on the GC to visualize it, and show on a table of values what happens

  5. x y -5.00000 -9.00000 -4.00000 -7.00000 -3.00000 undefined -2.00000 -3.00000 -1.00000 -1.00000 0.00000 1.00000 1.00000 3.00000 2.00000 5.00000 3.00000 7.00000 4.00000 9.00000 5.00000 11.00000 (C) Restrictions in Division – Graphical Interpretation

  6. (D) Examples of Long Division with Quadratic Equations • ex 1. Divide 2x² + 7x + 3 by x + 3 • Conclusions to be made • (i) x + 3 is a factor of 2x² + 7x + 3 • (ii) x + 3 divides evenly into 2x² + 7x + 3 • (iii) when 2x² + 7x + 3 is divided by x + 3, there is no remainder • (iv) 2x² + 7x + 3 = (x + 3)(2x + 1) • (v) (2x² + 7x + 3)/(x + 3) = 2x + 1 where x ≠ 3 • Show on GC and make connections • (i) graph 2x² + 7x + 3 and see that x = -3 is a root • (ii) graph (2x² + 7x + 3)/(x + 3) and see that we get a linear function with a hole in the graph at x = -3 which we can compare to the restrictions of the rational expression and we can comment on why the graph is a line

  7. (D) Examples of Long Division with Quadratic Equations - Graphs

  8. (D) Examples of Long Division with Quadratic Equations • ex 2. Divide (2x² + 7x + 3) ÷ (x + 4) and we get 2x - 1 with a remainder of 4 • Conclusions to be made • (i) x + 4 is a not factor of 2x² + 7x + 3 • (ii) x + 4 does not divide evenly into 2x² + 7x + 3 • (iii) when 2x² + 7x + 3 is divided by x + 4, there is a remainder of 7 • (iv) 2x² + 7x + 3 = (x + 4)(2x - 1) + 7 • (v) (2x² + 7x + 3)/(x + 4) = 2x - 1 + 7/(x + 4) • Show on GC and make connections • (i) graph 2x² + 7x + 3 and see that x = -4 is not a root • (ii) graph (2x² + 7x + 3)/(x + 4) and see a linear function (2x - 1) with an asymptote in the graph at x = -4 which we can compare to the restrictions of the rational expression and we can comment on why the graph is a line

  9. (D) Examples of Long Division with Quadratic Equations - Graphs

  10. One other graphic and algebraic observation  both divisions in the previous 2 examples have produced a quotient of Q(x) = 2x – 1  which then has a significance (see graph) which is ???????? (D) Examples of Long Division with Quadratic Equations - Graphs

  11. (E) Examples of Long Division with Cubic Equations • Divide 3x3 + 13x² - 9x + 5 by x + 5 • conclusions to be made: - all 5 conclusions are equivalent and say mean the same thing • (i) x + 5 is a factor of 3x3 + 13x² - 9x + 5 • (ii) x + 5 divides evenly into 3x3 + 13x² - 9x + 5 • (iii) when 3x3 + 13x² - 9x + 5 is divided by x + 5, there is no remainder • (iv) 3x3 + 13x² - 9x + 5 = (x + 5)(3x² - 2x + 1) • (v) (3x3 + 13x² - 9x + 5 )/(x + 5) = 3x² - 2x + 1 • Show on GC and make connections • i) graph 3x3 + 13x² - 9x + 5 and see that x = -5 is a root or a zero or an x-intercept • ii) graph (3x3 + 13x² - 9x + 5 )/(x + 5) and see a parabola has a hole in the graph at x = -5 which we can compare to the restrictions of the rational expression and we can comment on why the graph is a parabola.

  12. (E) Examples of Long Division with Cubic Equations - Graphs

  13. (E) Examples of Long Division with Cubic Equations • ex 3. Divide x3 - 42x + 30 by x - 6 show on GC and make connections • ex 4. Divide x2 + 6x3 - 5 by 2x - 1 show on GC and make connections • ex 5 Divide x4 + 4x3 + 2x² - 3x - 50 by x - 2 show on GC and make connections

  14. (E) Synthetic Division • Show examples 1,2,3 using both division methods • ex 3. Divide x3 - 42x + 30 by x - 6 • ex 4. Divide x2 + 6x3 - 5 by 2x - 1 • ex 5 Divide x4 + 4x3 + 2x² - 3x - 50 by x - 2 • Follow this link for some reading and review of synthetic division of polynomials from Steve Mayer at Bournemouth and Poole College

  15. (F) Homework • Nelson text page 43, Q3eol,4eol,8eol,9eol,10-12

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