Identify the terms and find their factors for 2x³ + 3x² + 4x + 1
Answers
Answer:
There is no easy factorization here.
There is a method to solve a cubic equation in general by hand (and calculator) on paper.
This method is based on the substitution of Vieta method.
Dividing by the first coefficient yields :
x
3
−
(
3
2
)
x
2
−
2
x
−
2
=
0
Substituting x=y+p in
x
3
+
a
x
2
+
b
x
+
c
=
0
yields :
y
3
+
(
3
p
+
a
)
y
2
+
(
3
p
2
+
2
a
p
+
b
)
y
+
p
3
+
a
p
2
+
b
p
+
c
=
0
if we take 3p+a=0 or p=-a/3, the first coefficient becomes zero, and we get :
y
3
−
(
11
4
)
y
−
(
13
4
)
=
0
(with p = 1/2)
Substituting y=qz in
y
3
+
b
y
+
c
=
0
, yields :
z
3
+
b
z
q
2
+
c
q
3
=
0
if we take q = sqrt(|b|/3), the coefficient of z becomes 3 or -3, and we get :
(here q = 0.95742711)
z
3
−
3
z
−
3.70310650
=
0
Substituting z = t + 1/t, yields :
t
3
+
1
t
3
−
3.70310650
=
0
Substituting
u
=
t
3
, yields the quadratic equation :
u
2
−
3.70310650
u
+
1
=
0
A root of this quadratic equation is u=3.40983738.
Substituting the variables back, yields :
t = cuberoot(u) = 1.50514344.
z = 2.16953194.
y = 2.07716869.
x
=
2.57716869
.
The other roots can be found by dividing and solving the remaining quadratic equation.
The other roots are complex :
−
0.53858435
±
0.69711716
i
.