We'll find the interpolating polynomial passing through the points
(
1
,
−
6
)
{\displaystyle (1,-6)}
(
2
,
2
)
{\displaystyle (2,2)}
(
4
,
12
)
{\displaystyle (4,12)}
Lagrange method .
We first use the formula to write the following:
p
(
x
)
=
−
6
(
x
−
2
)
(
1
−
2
)
(
x
−
4
)
(
1
−
4
)
+
2
(
x
−
1
)
(
2
−
1
)
(
x
−
4
)
(
2
−
4
)
+
12
(
x
−
1
)
(
4
−
1
)
(
x
−
2
)
(
4
−
2
)
{\displaystyle p(x)=-6{\frac {(x-2)}{(1-2)}}{\frac {(x-4)}{(1-4)}}+2{\frac {(x-1)}{(2-1)}}{\frac {(x-4)}{(2-4)}}+12{\frac {(x-1)}{(4-1)}}{\frac {(x-2)}{(4-2)}}}
After some simplification, we get:
p
(
x
)
=
−
2
(
x
−
2
)
(
x
−
4
)
−
1
(
x
−
1
)
(
x
−
4
)
+
2
(
x
−
1
)
(
x
−
2
)
{\displaystyle p(x)=-2(x-2)(x-4)-1(x-1)(x-4)+2(x-1)(x-2)}
p
(
x
)
=
−
2
(
x
2
−
6
x
+
8
)
−
1
(
x
2
−
5
x
+
4
)
+
2
(
x
2
−
3
x
+
2
)
{\displaystyle p(x)=-2(x^{2}-6x+8)-1(x^{2}-5x+4)+2(x^{2}-3x+2)}
And our answer:
p
(
x
)
=
−
x
2
+
11
x
−
16
{\displaystyle p(x)=-x^{2}+11x-16}
Now we'll add a point to our data set, and find a new interpolating polynomial. Let us add the point
(
3
,
−
10
)
{\displaystyle (3,-10)}
p
(
x
)
=
−
6
(
x
−
2
)
(
1
−
2
)
(
x
−
4
)
(
1
−
4
)
(
x
−
3
)
(
1
−
3
)
+
2
(
x
−
1
)
(
2
−
1
)
(
x
−
4
)
(
2
−
4
)
(
x
−
3
)
(
2
−
3
)
+
12
(
x
−
1
)
(
4
−
1
)
(
x
−
2
)
(
4
−
2
)
(
x
−
3
)
(
4
−
3
)
−
10
(
x
−
1
)
(
3
−
1
)
(
x
−
2
)
(
3
−
2
)
(
x
−
4
)
(
3
−
4
)
{\displaystyle p(x)=-6{\frac {(x-2)}{(1-2)}}{\frac {(x-4)}{(1-4)}}{\frac {(x-3)}{(1-3)}}+2{\frac {(x-1)}{(2-1)}}{\frac {(x-4)}{(2-4)}}{\frac {(x-3)}{(2-3)}}+12{\frac {(x-1)}{(4-1)}}{\frac {(x-2)}{(4-2)}}{\frac {(x-3)}{(4-3)}}-10{\frac {(x-1)}{(3-1)}}{\frac {(x-2)}{(3-2)}}{\frac {(x-4)}{(3-4)}}}
Simplifying, we get:
p
(
x
)
=
(
x
−
2
)
(
x
−
4
)
(
x
−
3
)
+
(
x
−
1
)
(
x
−
4
)
(
x
−
3
)
+
2
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
+
5
(
x
−
1
)
(
x
−
2
)
(
x
−
4
)
{\displaystyle p(x)=(x-2)(x-4)(x-3)+(x-1)(x-4)(x-3)+2(x-1)(x-2)(x-3)+5(x-1)(x-2)(x-4)}
p
(
x
)
=
(
x
3
−
9
x
2
+
26
x
−
24
)
+
(
x
3
−
8
x
2
+
19
x
−
12
)
+
2
(
x
3
−
6
x
2
+
11
x
−
6
)
+
5
(
x
3
−
7
x
2
+
14
x
−
8
)
{\displaystyle p(x)=(x^{3}-9x^{2}+26x-24)+(x^{3}-8x^{2}+19x-12)+2(x^{3}-6x^{2}+11x-6)+5(x^{3}-7x^{2}+14x-8)}
And our polynomial is:
p
(
x
)
=
9
x
3
−
64
x
2
+
137
x
−
88
{\displaystyle p(x)=9x^{3}-64x^{2}+137x-88}
