S5_Unit_2_Outcome_1 - Higher Unit 2 Higher Outcome 1 What...

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www.mathsrevision.com Higher Outcome 1 Higher Unit 2 Higher Unit 2 What is a polynomials Evaluating / Nested / Synthetic Method Factor Theorem Factorising higher Orders Finding Missing Coefficients Finding Polynomials from its zeros Factors of the form (ax + b) Credit Quadratic Theory Discriminant Condition for Tangency Completing the square
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www.mathsrevision.com Higher Outcome 1 Polynomials Definition A polynomial is an expression with several terms. These will usually be different powers of a particular letter. The degree of the polynomial is the highest power that appears. Examples 3x 4  – 5x 3  + 6x 2  – 7x - 4 Polynomial in x of degree 4. 7m 8  – 5m 5  – 9m 2  + 2 Polynomial in m of degree 8. w 13  – 6 Polynomial in w of degree 13. NB : It is not essential to have all the powers from the highest down, however powers  should be in descending order.
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www.mathsrevision.com Higher Outcome 1 Coefficients Disguised Polynomials (x + 3)(x – 5)(x + 5) = (x + 3)(x 2  – 25) =   x 3  + 3x 2  – 25x - 75 So this is a polynomial in x of degree 3. In the polynomial     3x 4  – 5x 3  + 6x 2  – 7x – 4 we say that   the coefficient of   x 4    is   3 the coefficient of   x 3    is   -5 the coefficient of   x 2    is   6 the coefficient of   x    is   -7 and the coefficient of   x 0    is   -4 (NB: x 0  = 1) In   w 13  – 6 , the coefficients of  w 12 , w 11 , ….w 2 , w are all zero.
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www.mathsrevision.com Higher Outcome 1 Evaluating Polynomials Suppose that  g(x)   =  2x 3  - 4x 2  + 5x - 9 Substitution Method g(2) = (2  X  2  X  2  X  2) – (4  X  2  X  2 ) + (5  X  2) - 9 =  16 – 16 + 10 - 9 =   1 NB :    this requires 9 calculations.
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www.mathsrevision.com Higher Outcome 1 Nested or Synthetic Method This involves using the coefficients and requires fewer calculations so is more  efficient. It can also be carried out quite easily using a calculator. g(x)   =  2x 3  - 4x 2  + 5x - 9 Coefficients are 2,    -4,      5,    -9 g(2)  =    2 -4 5        -9 2 4 0 0 5 10 1 This requires only 6 calculations so is  1 / 3  more efficient.
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www.mathsrevision.com Higher Outcome 1 Example If f(x)   =  2x 3  - 8x then the coefficients are 2 0 -8 0 and f(2)   =   2 2 0 -8  0 2 4 4 8 0 0 0 Nested or Synthetic Method
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www.mathsrevision.com Higher Outcome 1 Factor Theorem If (x – a)  is a factor of the polynomial  f(x) Then f(a) = 0. Reason Say   f(x)  =  a 3 x 3  + a 2 x 2  + a 1 x + a 0  =  (x – a)(x – b)(x – c) polynomial form        factorised form Since (x – a), (x – b) and (x – c) are factors  then  f(a) = f(b) = f(c ) = 0 Check           f(b) = (b – a)(b – b)(b – c) = (b – a)  X  0  X  (b – c) = 0
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This note was uploaded on 02/13/2012 for the course MAT 205 math 205 taught by Professor Google during the Spring '10 term at University of Phoenix.

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S5_Unit_2_Outcome_1 - Higher Unit 2 Higher Outcome 1 What...

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