The polynomial y = - 3 · x³ - 2 · x² + x is the equation that passes through given points.
How to determine the polynomial equation that passes through a given set of points
According to algebra, a polynomial of grade n can be found by knowning n + 1 distinct points of the form (x, y). Since we need to determine a polynomial grade 3, then we need four distinct points and definition of polynomial:
y = a · x³ + b · x² + c · x + d, where a, b, c, d are real coefficients.
If we know that (8, - 1656), (- 2, 14), (9, - 2340) and (10, - 3190), then the coefficients of the polynomials are:
a · 8³ + b · 8² + c · 8 + d = - 1656
a · (- 2)³ + b · (- 2)² + c · (- 2) + d = 14
a · 9³ + b · 9² + c · 9 + d = - 2340
a · 10³ + b · 10² + c · 10 + d = - 3190
The resulting system of linear equations is:
512 · a + 64 · b + 8 · c + d = - 1656
- 8 · a + 4 · b - 2 · c + d = 14
729 · a + 81 · b + 9 · c + d = - 2340
1000 · a + 100 · b + 10 · c + d = - 3190
Then, the resulting coefficients of the polynomial are:
(a, b, c, d) = (- 3, - 2, 1, 0)
The polynomial that passes through given points is y = - 3 · x³ - 2 · x² + x.