Grade 11 Analysis and Approaches HL only
The Remainder Theorem.
1. Long Division:
a) Determine (4 x 3  10 x 2  6 x  18)  (2 x  5) , write the division statement and state any
restrictions
b) When a certain polynomial is divided by x + 3 , the quotient is x 2  3 x  5 and the remainder
is 6. What is the polynomial?
2.:
(a)
(b)
(c)
(d)
Divide f ( x )  x 3  x 2  7 by ( x  2 )
What is the remainder?
What is the value of f (2 ) ?
What is the relationship between the remainder and the value of f (2 ) ?
The Remainder Theorem:
When a polynomial function P(x) is divided by x  b , the remainder is P(b); and when it
b
is divided by ax  b , the remainder is P   , where a and b are integers, and a ≠ 0.
a
Ex 1:
Without dividing, determine the remainder when f ( x )  2 x 3  4 x 2  3 x  6 is
divided by  x  2 
Ex 2:
Without dividing, determine the remainder when g ( x)  2 x 3  x 2  3 x  6 is
divided by 2 x  3
Ex 3:
When x 3  3 x 2  kx  10 is divided by  x  5  , the remainder is 15. Determine
the value of k.
Ex 4:
When f ( x)  ax 3  bx 2  x  3 is divided by  x  1 , the remainder is 4. If
f ( x ) is divided by  x  2  , the remainder is –47. What are the values of a and b?
The Factor Theorem:
( x  b ) is a factor of f ( x ) if and only if f (b)  0
b
Similarly, ax  b is a factor of f ( x ) if and only if f    0
a
FACTORING A POLYNOMIAL using the FACTOR THEOREM
(i) Determine one zero, b, of the polynomial f ( x ) i.e.
f (b )  0
(ii) Divide f ( x ) by  x  b  to find another factor
(iii) If the quotient can be factored further, continue factoring.
(iv) State the factored form of the polynomial f ( x ) .
2) Factor the given polynomials:
a) x 3  3 x 2  13 x  15
b) 4 x 3  8 x 2  x  2
c) 2 x 3  9 x 2  7 x  6
d) x 4  3x 3  7 x 2  27 x 18
Exercise 5G
Answers