The Factor Theorem Flashcards

1
Q

Content for factor theorem:

A
  • Given polynomial f ( k ), and a constant k
  • If f ( k ) = 0, then ( x - k ) is the factor of f ( k )
  • ( If f ( 1 ) = 0, then ( x - 1 ) is the factor of f ( 1 ) )
  • If ( x - k ) is a factor of f ( k ), then f ( k ) = 0
  • ( If ( x - 1 ) is a factor of f ( k ), then f ( 1 ) = 0 )

Steps:

  • Substitute values into the polynomial until you find f ( k ) = 0
  • Divide the polynomial by ( x - k ) to find the quadratic
  • Write f ( k ) = ( x - k ) ( ax^2 + bx + c )
  • Factorise the quadratic to have it in linear factors ( if possible )
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2
Q
Show that ( x - 2 ) is a factor of p( x ) = x^3 + x^2 - 4x - 4.
Hence factorise p( x ) fully.
A
  • ( x - 2 ) so x = 2
  • ( Substitute x = 2 into equation and it should = 0 )
  • x - 2 | x^3 + x^2 - 4x - 4 | ( Dividing )
  • Quadratic = x^2 + 3x + 2
  • ( Factorise )
  • ( x + 1 ) ( x + 2 )
  • p( x ) = ( x - 2 ) ( x + 1 ) ( x + 2 )
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3
Q

Fully factorise f( x ) = 2x^3 - 3x^2 + x - 6

A
  • ( Firstly find a value which makes f( x ) = 0 )
  • When x = 2, f( x ) = 0
  • So ( x - 2 ) is a factor
  • x - 2 | 2x^3 - 3x^2 + x - 6 | ( Dividing )
  • ( 2x^2 + x + 3 ) ( x - 2 )
  • ( Not possible to factorise )
  • Hence f( x ) = ( 2x^2 + x + 3 ) ( x - 2 )
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