Question

Find the cubic equation:

f(x) = ax^3+bx^2+cx+d

for which f(-1)=3, f(1)=1, f(2)=6, and f(3)=7.

Find the value of a, b, c, and d

Answer 1

Augmented matrix for given system of equations

Your matrix

	X1	X2	X3	X4	b
1	-1	1	-1	1	3
2	1	1	1	1	1
3	8	4	2	1	6
4	27	9	3	1	7

Find the pivot in the 1st column in the 1st row (inversing the sign in the whole row)

	X1	X2	X3	X4	b
1	1	-1	1	-1	-3
2	1	1	1	1	1
3	8	4	2	1	6
4	27	9	3	1	7

Eliminate the 1st column

	X1	X2	X3	X4	b
1	1	-1	1	-1	-3
2	0	2	0	2	4
3	0	12	-6	9	30
4	0	36	-24	28	88

Make the pivot in the 2nd column by dividing the 2nd row by 2

	X1	X2	X3	X4	b
1	1	-1	1	-1	-3
2	0	1	0	1	2
3	0	12	-6	9	30
4	0	36	-24	28	88

Eliminate the 2nd column

	X1	X2	X3	X4	b
1	1	0	1	0	-1
2	0	1	0	1	2
3	0	0	-6	-3	6
4	0	0	-24	-8	16

Make the pivot in the 3rd column by dividing the 3rd row by -6

	X1	X2	X3	X4	b
1	1	0	1	0	-1
2	0	1	0	1	2
3	0	0	1	1/2	-1
4	0	0	-24	-8	16

Eliminate the 3rd column

	X1	X2	X3	X4	b
1	1	0	0	-1/2	0
2	0	1	0	1	2
3	0	0	1	1/2	-1
4	0	0	0	4	-8

Make the pivot in the 4th column by dividing the 4th row by 4

	X1	X2	X3	X4	b
1	1	0	0	-1/2	0
2	0	1	0	1	2
3	0	0	1	1/2	-1
4	0	0	0	1	-2

Eliminate the 4th column

	X1	X2	X3	X4	b
1	1	0	0	0	-1
2	0	1	0	0	4
3	0	0	1	0	0
4	0	0	0	1	-2

Solution set:

a = -1

b = 4

c = 0

d = -2