# Hotmath

Show that the
*
n
*
th–order differences for the given function of degree
*
n
*
are nonzero and constant:

*
f
*
(
*
x
*
) = –
*
x
*
^{
3
}
+ 4
*
x
*
^{
2
}
– 3
*
x
*
– 4

Show that the
*
n
*
th–order differences for the given function of degree
*
n
*
are nonzero and constant:

*
f
*
(
*
x
*
) = 4
*
x
*
^{
4
}
– 40
*
x
*

A formula for the
*
n
*
th triangular number is
*
f
*
(
*
n
*
) = 1/2(
*
n
*
^{
2
}
+
*
n
*
).

Show that this function has constant second–order differences.

What is the degree of a polynomial
*
P
*
(
*
x
*
)
*
*
if, in the equation

*
y
*
=
*
P
*
(
*
x
*
),

the
*
y
*
–values are all equal for the 11
^{
th
}
set of differences of consecutive
*
x
*
–values, and not equal for the 10
^{
th
}
set of differences?

Find the values of the first differences of the function defined by given data points and find the degree of the function.

Plot the data points and find an equation for the line passing through the plotted points.

For the given pattern, find
*
f
*
(5) and
*
f
*
(6).

*
f
*
(1) = 1
^{
3
}
=1

*
f
*
(2) = 1
^{
3
}
+ 2
^{
3
}
= 9

*
f
*
(3) = 1
^{
3
}
+ 2
^{
3
}
+ 3
^{
3
}
= 36

*
f
*
(4) = 1
^{
3
}
+ 2
^{
3
}
+ 3
^{
3
}
+ 4
^{
3
}
= 100

Also find the degree of the polynomial
*
f
*
(n) using the Polynomial –Difference Theorem.

Say whether the following data can be modeled by a polynomial function. If it is polynomial then find its degree.

Determine the degree of the polynomial function that models the data given in the table below.

Find a formula for the polynomial function.