1.5 #30.

Find the cubic that goes through (-2,2), (-1,2), (1,2), (2, 10).

Solution: Let p(x) = ax^3 + bx^2 + cx + d. Then:

p(-2)=2  ==>  -8a + 4b - 2c + d = 2

p(-1)=2  ==>   -a +  b -  c + d = 2

p(1) =2  ==>    a +  b +  c + d = 2

p(2)=10  ==>   8a + 4b + 2c + d = 10

or,

[ -8  4  -2  1  ][a]  [ 2]
[ -1  1  -1  1  ][b] =[ 2] 
[  1  1   1  1  ][c]  [ 2]
[  8  4   2  1  ][d]  [10]

rref gives:

[1  0  0  0  2/3]
[0  1  0  0  4/3]
[0  0  1  0 -2/3]
[0  0  0  1  2/3].

Thus, p(x) = (2x^3 + 4x^2 - 2x + 2)/3.