Factor and Remainder Theorems
Notes:
Factor Theorem: (x – a) is a factor of a polynomial f(x) if f(a) = 0.
Remainder Theorem: The remainder when a polynomial f(x) is divided by (x – a) is f(a).
Extended version of the factor theorem:
 b 
(ax + b) is a factor of a polynomial f(x) if f    0 .
 a 
Example:
Show that (x – 3) is a factor of x3  2 x 2  5 x  6 and find the other two factors.
Sketch the graph of y = x3  2 x 2  5 x  6 , showing the points where it cuts the x and y axes.
Solve the inequality x3  2 x 2  5 x  6 > 0.
Solution:
Let f(x) = x3  2 x 2  5 x  6 .
(x – 3) is a factor if f(3) = 0.
f(3) = 33  2  32  5  3  6  27  18  15  6  0 .
So (x – 3) is a factor.
To find the other factors we divide x3  2 x 2  5 x  6 by (x – 3):
x2  x  2
x  3 x3  2 x 2  5 x  6
x3  3x 2
x2  5x
x 2  3x
 2x  6
2 x  6
0
To find the other factors we have to factorise x 2  x  2  ( x  1)( x  2)
So x3  2 x 2  5 x  6 =(x – 3)(x – 1)(x + 2).
y
10
From the graph we see that x3  2 x 2  5 x  6 > 0 if:
x > 3 OR -2 < x < 1
5
-4
-3
-2
-1
1
-5
-10
2
3
4
5
x
Example 2:
Find the remainder when 2 x3  5 x 2  2 x  7 is divided by (x + 2).
Solution:
Let f(x) = 2 x3  5 x 2  2 x  7
The remainder when f(x) is divided by (x + 2) is f(-2).
f(-2) = 2(2)3  5(2)2  2(2)  7  16  20  4  7  7 .
Revision Questions:
1.
Show that (x – 2) is a factor of P(x), where
P(x) = x3  3 x 2  10 x  24 ,
and find the other two factors.
(a) Show that (x – 3) is a factor of x3  x 2  8 x  12 and find the other two factors.
(b) Sketch the graph of y = x3  x 2  8 x  12 .
(c) Solve the inequality x3  x 2  8 x  12 > 0.
2.
3.
When 2 x3  x 2  13x  k is divided by x – 2 the remainder is -20. Show that k = -6.
4.
(i) Multiply x 2  3 x  5 by 2 x 2  x  4 .
(ii) Find values of p, q, r, s such that
(2 x  3)( px3  qx 2  rx  s)  2 x 4  13x3  7 x 2  18
for all values of x.
(iii) Simplify ( x  1)(3x 2  2 x  5)  ( x  1)(2 x 2  x  1) as the product of three factors.
5.
P(x) = 2 x3  9 x 2  x  12 .
(a) Use the factor theorem to find a factor (x – a) of P(x), where a is an integer.
(b) Find the other two factors.
(c) Sketch the graph of y = P(x), marking the coordinates of the points where it meets the xaxis and y-axis.
(d) Solve P(x) > 0.
6.
(a)
(b)
(c)
(d)
g(x) = x3  3x 2  13x  15 .
Show that g(-5) = 0 and g(3) = 0.
Hence factorise g(x).
Sketch the graph of y = g(x).
Hence write down the full set of values of x for which g(x) > 0.
7.
(a) Factorise x3  4 x 2  x  6 , given that (x – 2) is a factor.
(b) Solve the inequality x3  4 x 2  x  6  2( x  3)( x  1).
8.
Find the quotient and the remainder when x3  2 x 2  x  3 is divided by (x – 4).
[Hint: Use long division to divide x3  2 x 2  x  3 by (x – 4)].
9.
Factorise x3  4 x 2  x  6 , given that (x – 2) is a factor.
Solve the inequality x3  4 x2  x  6  2( x  3)( x  1)