Polynomials

• synthetic division


  used to factorise a polynomial

  anything more than $x^2$

  $x^2$= a quadratic
• difference between a factor and a root


  x+5 is a factor

  x=-5 is a root
• how to do synthetic division


  1. detach the coefficient and write them across the top row of the table (MUST BE IN DECREASING DEGREE)
  2. write the number for which you want to evaluate the polynomial (root or whats inside the bracket of a function)
  3. bring down the first coefficient
  4. multiply this by the input and write it under the next coefficient
  5. add the numbers in that column
  6. repeat
  7. number in the lower right cell is the remainder
• show that *(x-4)* is a factor


  • MUST WRITE SINCE REMAINDER IS ZERO, X-4 IS A FACTOR
  • use factor to find root
  • put it on the left of the table
  • synthetic division TO SHOW REMAINDER IS 0
• ... to evaluate f(-2), given f(x)....


  • do synthetic division
  • write the answer as f(-2)=[remainder]
• quotient 


  answer you get at the bottom of the table after synthetic division
• fully factorise...


  • polynomial=synthetic division
  • use the factors of the whole number in the equation to put as the divider (on the left) of the table
  • keep trying until remainder=0
  • factorise using that quotient and the factor of the divider (factor of the divider)(quotient)

  e.g $(x-3)(2x^2+11x+5)$

  $(x-3)(2x+1)(x+5)$
• find the solution of...

  means final form should be one or more x=

  • synthetic division using factors of the whole number in the equation
  • when remainder=0, turn the root/divider into a factor and factorise using that and the quotient
  • solve to find x
• .... and express f(x) as (x+1)q(x)+f(h)


  f(h) = remainder

  -use factor in the question and turn it into a root to use as the divider

  -synthetic division

  -write answer as (factor)(quocient) -/+ remainder
• finding unknown coefficients


  * with a factor, the remainder will always equal 0*

  -use factor to turn into root to use as the divider

  -use synthetic division putting in the coefficient where appropriate

  e.g $px$ -> would write $p$ at the top for the coefficient of x

  -WRITE SINCE REMAINDER=0

  • use remainder from division to solve for p
• if it tells you the remainder, remember to set the remainder to that and not 0
• 2 unknowns=


  similtanious equations
• finding intersections of curves(quadratic)

  1. set equations equal to each other
  2. move all to one side and set equal to 0
  3. factorise
  4. find x
  5. substitute x into line equation to find y
  6. write coordinates
  7. repeat if it has two roots (result in two x answers for step 4)
• finding intersections of curves (polynomials)


  1. set equal to each other
  2. move to one side and set equal to 0
  3. use factors of the whole number in the equation as divider
  4. keep going until remainder = 0
  5. make divider a factor

  WRITE X=*1 IS A ROOT SO X-*1 IS A FACTOR

  6. write factor and quocient in brackets

  factorise and solve to find x's

  7. substitute one at a time x into the line equation to find y

  8. write co ordinate
• Determining the equation of a curve

  1. write out the roots and rearrange to get the factors
  2. write these factors out in an equation equal to y and multiplied together by k

  $y=$$k(x+a)(x+b)(x+c)$

  3. substitute the coordinates of a known point into this equation and solve to find k

  4. replace k with this value and rewrite the equation back into a polynomial
• repeated roots


  if a repeated root exists, then a stationary point lies on the x-axis

  repeated roots exist when two roots, and hence two factors, are equal

  (touches the x-axis but doesnt change from + to -