- Source: Trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
Examples of trinomial expressions
3
x
+
5
y
+
8
z
{\displaystyle 3x+5y+8z}
with
x
,
y
,
z
{\displaystyle x,y,z}
variables
3
t
+
9
s
2
+
3
y
3
{\displaystyle 3t+9s^{2}+3y^{3}}
with
t
,
s
,
y
{\displaystyle t,s,y}
variables
3
t
s
+
9
t
+
5
s
{\displaystyle 3ts+9t+5s}
with
t
,
s
{\displaystyle t,s}
variables
a
x
2
+
b
x
+
c
{\displaystyle ax^{2}+bx+c}
, the quadratic polynomial in standard form with
a
,
b
,
c
{\displaystyle a,b,c}
variables.
A
x
a
y
b
z
c
+
B
t
+
C
s
{\displaystyle Ax^{a}y^{b}z^{c}+Bt+Cs}
with
x
,
y
,
z
,
t
,
s
{\displaystyle x,y,z,t,s}
variables,
a
,
b
,
c
{\displaystyle a,b,c}
nonnegative integers and
A
,
B
,
C
{\displaystyle A,B,C}
any constants.
P
x
a
+
Q
x
b
+
R
x
c
{\displaystyle Px^{a}+Qx^{b}+Rx^{c}}
where
x
{\displaystyle x}
is variable and constants
a
,
b
,
c
{\displaystyle a,b,c}
are nonnegative integers and
P
,
Q
,
R
{\displaystyle P,Q,R}
any constants.
Trinomial equation
A trinomial equation is a polynomial equation involving three terms. An example is the equation
x
=
q
+
x
m
{\displaystyle x=q+x^{m}}
studied by Johann Heinrich Lambert in the 18th century.
= Some notable trinomials
=The quadratic trinomial in standard form (as from above):
a
x
2
+
b
x
+
c
{\displaystyle ax^{2}+bx+c}
sum or difference of two cubes:
a
3
±
b
3
=
(
a
±
b
)
(
a
2
∓
a
b
+
b
2
)
{\displaystyle a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})}
A special type of trinomial can be factored in a manner similar to quadratics since it can be viewed as a quadratic in a new variable (xn below). This form is factored as:
x
2
n
+
r
x
n
+
s
=
(
x
n
+
a
1
)
(
x
n
+
a
2
)
,
{\displaystyle x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),}
where
a
1
+
a
2
=
r
a
1
⋅
a
2
=
s
.
{\displaystyle {\begin{aligned}a_{1}+a_{2}&=r\\a_{1}\cdot a_{2}&=s.\end{aligned}}}
For instance, the polynomial x2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a1 = −2 and a2 = −1 of the above system gives the trinomial factorization:
x2 + 3x + 2 = (x + a1)(x + a2) = (x + 2)(x + 1).
The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
See also
Notes
References
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