Answer:
to do...
Look at each possible power of
x
in descending order and add up the different ways of getting it.
So in our example:
Given:
(
x
+
1
)
(
x
2
+
x
+
1
)
we can tell that the highest possible power of
x
in the product is
3
, so work down from there:
x
3
: This can only result from multiplying the
x
in the binomial by the
x
2
in the trinomial, so the coefficient is:
1
⋅
1
=
1
So we can start to write:
(
x
+
1
)
(
x
2
+
x
+
1
)
=
x
3
...
x
2
: This can result from
x
⋅
x
or
1
⋅
x
2
, so the coefficient is:
1
⋅
1
+
1
⋅
1
=
2
So we can add
+
2
x
2
to the result:
(
x
+
1
)
(
x
2
+
x
+
1
)
=
x
3
+
2
x
2
...
x
1
:
This can result from
x
⋅
1
or
1
⋅
x
, so the coefficient is:
1
⋅
1
+
1
⋅
1
=
2
So we can add
+
2
x
to the result:
(
x
+
1
)
(
x
2
+
x
+
1
)
=
x
3
+
2
x
2
+
2
x
...
x
0
:
The constant term can only result from multiplying the constant term of the binomial by that of the trinomial, so:
1
⋅
1
=
1
So our final result is:
(
x
+
1
)
(
x
2
+
x
+
1
)
=
x
3
+
2
x
2
+
2
x
+
1
In practice (and with practice) the result line is all you need write: Adding up the coefficients can be done in your head
