Example
x3 + 3x2 – 10x - 24
Factorise
We need some trial &
error with factors of –24
ie
+/-1, +/-2, +/-3 etc
1
f(-1) = -1
3
-1
2
1
f(1) = 1
-10
-2
-12
-24
12
-12
1
3
1
-10
4
-24
-6
1
4
-6
-30
No good
No good
f(-2) = -2 1
1
3
-2
1
Other factor is
So
-10
-2
-12
-24
24
0
f(-2) = 0
so (x + 2) a factor
x2 + x - 12 = (x + 4)(x – 3)
x3 + 3x2 – 10x – 24 = (x + 4)(x + 2)(x – 3)
Roots/Zeros
The roots or zeros of a polynomial tell us where it cuts
the X-axis. ie where f(x) = 0.
If a cubic polynomial has zeros a, b & c then it has
factors (x – a), (x – b) and (x – c).
Example
Solve
x4 + 2x3 - 8x2 – 18x – 9 = 0
We need some trial &
error with factors of –9
ie
+/-1, +/-3 etc
f(-1) = -1
1
1
2
-1
1
-8
-1
-9
-18
9
-9
-9
9
0
f(-1) = 0
so (x + 1) a
factor
Other factor is x3 + x2 – 9x - 9 which we can call g(x)
test
+/-1, +/-3 etc
g(-1) = -1 1
1
1
-1
0
-9
0
-9
-9
9
0
g(-1) = 0
so (x + 1) a
factor
Other factor is x2 – 9 = (x + 3)(x – 3)
if
x4 + 2x3 - 8x2 – 18x – 9 = 0
then
(x + 3)(x + 1)(x + 1)(x – 3) = 0
So
x = -3 or x = -1 or x = 3
Breakdowns
ax3 + bx2 + cx + d
A cubic polynomial ie
could be factorised into either
(i) Three linear factors of the form (x + a) or (ax + b)
or
(ii) A linear factor of the form (x + a) or (ax + b) and a
.
quadratic factor (ax2 + bx + c) which doesn’t factorise.
or
(iii) It may be irreducible.
IT DIZNAE
FACTORISE