1.4 - Dividing Polynomials

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1.4 - Dividing Polynomials

• MCB4U

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(A) Review

• recall the steps involved in long division
• set it up using the example of 30498 39

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(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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(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

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(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

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(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

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

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

• 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
  ????????

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(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.

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

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(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

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(F) Homework

• Nelson text page 43, Q3eol,4eol,8eol,9eol,10-12