Find the cubic polynomial whose graph passes through the points ( -1, -1), (0, 1), (1, 3), (4, -1).

Emily-Jane Bray

Emily-Jane Bray

Answered question

2021-02-14

Find the cubic polynomial whose graph passes through the points (1,1),(0,1),(1,3),(4,1).

Answer & Explanation

Macsen Nixon

Macsen Nixon

Skilled2021-02-15Added 117 answers

Step 1
Denote the interpolating cubic polynomial by
p(x)=a0+a1x+a2x2+a3x3
Let's substitute the given points in the equation of the polynomial. The points (1,1),(0,1),(1,3)and(4,1) thus have to satisfy.
1=a0a1+a2a3
1=a0
3=a0+a1+a2+a3
1=a0+4a1+16a2+64a3
First, let's substitute a0=1 into the three remaining equations. That way we reduced the original system to a system of three equations with three unknowns:
a1+a2a3=2
a1+a2+a3=2
4a1+16a2+64a3=2
Let's solve this system using Gauss-Jordan elimination. The augmented matrix of the system is.
[11121112416642]
Step 2
Multiply the first row by -1.
[11121112416642]
Add -1 times the first row to the second row, Add -4 times the first row to the third row.
[111202000206010]
Multiply the second row by 12
[111201000206010]
Add -20 times the second row to the third row.
[11120100006010]
Multiply the third row by 160.
[111201000011/6]
Add -1 times the third row to the first row.
[11013/601000011/6]
Add 1 times the second row to the first row.
[10013/601000011/6]
The solution is thus
=136,a2=0,a3=16

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