Find the function fx ax3 bx2 cx d for which f3 135 f1

Find the function f(x) = ax^3 + bx^2 + cx + d for which f(-3) = -135, f(-1) = 1, f(1) = 9, and f(2) = 25. f(x) = (Simplify your answer.)

Solution

We get, on substituting the given x-values,one by one, in the equation , f(x)= ax³+bx²+cx+d, as under:

Now, let A be the augmented matrix of the above linear system. Then A =

-27

9

-3

1

-135

-1

1

-1

1

1

1

1

1

1

9

8

4

2

1

25

In order to solve the above linear system, we will reduce A to its RREF as under:

Multiply the 1st row by -1/27; Add 1 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row; Add -8 times the 1st row to the 4th row

Multiply the 2nd row by 3/2; Add -4/3 times the 2nd row to the 3rd row

Add -20/3 times the 2nd row to the 4th row; Multiply the 3rd row by 3/8

Add -10 times the 3rd row to the 4th row; Multiply the 4th row by -1/5

Add 1/3 times the 4th row to the 3rd row; Add -13/9 times the 4th row to the 2nd row

Add 1/27 times the 4th row to the 1st row; Add 4/3 times the 3rd row to the 2nd row

Add -1/9 times the 3rd row to the 1st row ; Add 1/3 times the 2nd row to the 1st row

Then the RREF of A is

1

0

0

0

4

0

1

0

0

-4

0

0

1

0

0

0

0

0

1

9

Hence, a = 4,b = -4, c = 0 and d = 9. Then, the required function is f(x)= 4x³-4x²+9.

-27	9	-3	1	-135
-1	1	-1	1	1
1	1	1	1	9
8	4	2	1	25