Download A Factor and Remainder Theorems

Term 1a
Pre U Topic Review
factor and remainder theorems
1. Use the Factor Theorem to show that:
a) ( x  1) is a factor of x3  2 x 2  11x  12 .
(2 marks)
b) ( x  3) is a factor of  x3  x 2  7 x  3 .
(2 marks)
c) (2 x  1) is a factor of 2 x3  x 2  2 x  1 .
(2 marks)
2. Use algebraic division to find the quotient and remainder (if any) when :
a) x3  2 x 2  3x  2 is divided by ( x  1) .
(3 marks)
b) 2 x3  4 x 2  5 x  3 is divided by ( x  2) .
(3 marks)
c) x 3  2 x 2  4 is divided by ( x  3) .
(3 marks)
3. In the following equations one solution is given. Factorise each expression and hence solve
the equation:
a) x3  6 x 2  11x  6  0 ; x  1 is one solution.
(3 marks)
b) 2 x3  x 2  5 x  2  0 ; x   12 is one solution.
(3 marks)
4. If ( x  2) is a factor of x3  2 x 2  ax  6 , find the value of a.
(3 marks)
5. Find the remainder when :
a) x 2  5 x  2 is divided by ( x  2) .
(2 marks)
b) 2 x3  x 2  5 x  1 is divided by (2 x  1) .
(3 marks)
6. f( x)  2 x3  5x 2  7 x  3 . Show that ( x  2) is not a factor of f( x) , and find the remainder
when f( x) is divided by (2 x  3) .
(5 marks)
7. The expression f( x)  x3  ax 2  bx  10 is divisible by ( x  2) but leaves a remainder of 12
when divided by ( x  1)
a) Find the values of the constants a and b.
b) Solve the equation f( x)  0 .
(6 marks)
(4 marks)
8. When x3  ax 2  2 x  1 is divided by ( x  2) the remainder is three times the remainder when
divided by ( x  1) . Find the value of a.
(6 marks)