The Factor Theorem

Question 1

Content for factor theorem:

Answer 1

• 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 )

Question 2

Show that ( x - 2 ) is a factor of p( x ) = x^3 + x^2 - 4x - 4.
Hence factorise p( x ) fully.

Answer 2

• ( 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 )

Question 3

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

Answer 3

• ( 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 )