1. 1.4 - Dividing Polynomials MCB4U

2. (A) Review recall the steps involved in long division: recall the steps involved in long division: set it up using the example of 30498 ÷ 39 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 sometimes it will be easier to factor a polynomial and simply cancel common factors ex. ex.

4. (C) Restrictions in Division any time we divide there is always one restriction, in that you cannot divide by zero 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 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. 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 With the example above, draw it on the GC to visualize it, and show on a table of values what happens

5. (C) Restrictions in Division – Graphical Interpretation 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

6. (D) Examples of Long Division with Quadratic Equations ex 1. Divide 2x² + 7x + 3 by x + 3 ex 1. Divide 2x² + 7x + 3 by x + 3 Conclusions to be made Conclusions to be made (i) x + 3 is a factor of 2x² + 7x + 3 (i) x + 3 is a factor of 2x² + 7x + 3 (ii) x + 3 divides evenly into 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 (iii) when 2x² + 7x + 3 is divided by x + 3, there is no remainder (iv) 2x² + 7x + 3 = (x + 3)(2x + 1) (iv) 2x² + 7x + 3 = (x + 3)(2x + 1) (v) (2x² + 7x + 3)/(x + 3) = 2x + 1 where x ≠ 3 (v) (2x² + 7x + 3)/(x + 3) = 2x + 1 where x ≠ 3 Show on GC and make connections Show on GC and make connections (i) graph 2x² + 7x + 3 and see that x = -3 is a root (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 (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 ex 2. Divide (2x² + 7x + 3) ÷ (x + 4) and we get 2x - 1 with a remainder of 4 Conclusions to be made Conclusions to be made (i) x + 4 is a not factor of 2x² + 7x + 3 (i) x + 4 is a not factor of 2x² + 7x + 3 (ii) x + 4 does not divide evenly into 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 (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 (iv) 2x² + 7x + 3 = (x + 4)(2x - 1) + 7 (v) (2x² + 7x + 3)/(x + 4) = 2x - 1 + 7/(x + 4) (v) (2x² + 7x + 3)/(x + 4) = 2x - 1 + 7/(x + 4) Show on GC and make connections Show on GC and make connections (i) graph 2x² + 7x + 3 and see that x = -4 is not a root (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 (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 ???????? 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 ????????

11. (E) Examples of Long Division with Cubic Equations Divide 3x 3 + 13x² - 9x + 5 by x + 5 Divide 3x 3 + 13x² - 9x + 5 by x + 5 conclusions to be made: - all 5 conclusions are equivalent and say mean the same thing conclusions to be made: - all 5 conclusions are equivalent and say mean the same thing (i) x + 5 is a factor of 3x 3 + 13x² - 9x + 5 (i) x + 5 is a factor of 3x 3 + 13x² - 9x + 5 (ii) x + 5 divides evenly into 3x 3 + 13x² - 9x + 5 (ii) x + 5 divides evenly into 3x 3 + 13x² - 9x + 5 (iii) when 3x 3 + 13x² - 9x + 5 is divided by x + 5, there is no remainder (iii) when 3x 3 + 13x² - 9x + 5 is divided by x + 5, there is no remainder (iv) 3x 3 + 13x² - 9x + 5 = (x + 5)(3x² - 2x + 1) (iv) 3x 3 + 13x² - 9x + 5 = (x + 5)(3x² - 2x + 1) (v) (3x 3 + 13x² - 9x + 5 )/(x + 5) = 3x² - 2x + 1 (v) (3x 3 + 13x² - 9x + 5 )/(x + 5) = 3x² - 2x + 1 Show on GC and make connections Show on GC and make connections i) graph 3x 3 + 13x² - 9x + 5 and see that x = -5 is a root or a zero or an x- intercept i) graph 3x 3 + 13x² - 9x + 5 and see that x = -5 is a root or a zero or an x- intercept ii) graph (3x 3 + 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. ii) graph (3x 3 + 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 x 3 - 42x + 30 by x - 6 show on GC and make connections ex 3. Divide x 3 - 42x + 30 by x - 6 show on GC and make connections ex 4. Divide x 2 + 6x 3 - 5 by 2x - 1 show on GC and make connections ex 4. Divide x 2 + 6x 3 - 5 by 2x - 1 show on GC and make connections ex 5 Divide x 4 + 4x 3 + 2x² - 3x - 50 by x - 2 show on GC and make connections ex 5 Divide x 4 + 4x 3 + 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 Show examples 1,2,3 using both division methods ex 3. Divide x 3 - 42x + 30 by x - 6 ex 3. Divide x 3 - 42x + 30 by x - 6 ex 4. Divide x 2 + 6x 3 - 5 by 2x - 1 ex 4. Divide x 2 + 6x 3 - 5 by 2x - 1 ex 5 Divide x 4 + 4x 3 + 2x² - 3x - 50 by x - 2 ex 5 Divide x 4 + 4x 3 + 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 Follow this link for some reading and review of synthetic division of polynomials from Steve Mayer at Bournemouth and Poole College Follow this link for some reading and review of synthetic division of polynomials from Steve Mayer Follow this link for some reading and review of synthetic division of polynomials from Steve Mayer

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