# The curve below is the graph of a degree 3 polynomial. It goes through the point (5,−31.5). How to find the polynomial ?

Nov 3, 2016

$y = - \frac{1}{2} {x}^{3} + {x}^{2} + 2 x - 4$

#### Explanation:

$y \left(x\right) = a {x}^{3} + b {x}^{2} + c x + d$

We can see that $y \left(0\right) = - 4$:

$y \left(x\right) = a {\left(0\right)}^{3} + b {\left(0\right)}^{2} + c \left(0\right) + d = - 4$

$d = - 4$

$y \left(x\right) = a {x}^{3} + b {x}^{2} + c x - 4$

We can see that $y \left(- 2\right) = 0$

$0 = a {\left(- 2\right)}^{3} + b {\left(- 2\right)}^{2} + c \left(- 2\right) - 4$

$- 8 a + 4 b - 2 c = 4$

We can see that $y \left(2\right) = 0$

$8 a + 4 b + 2 c = 4$

We are given the point $\left(5 , 31.5\right)$

$y \left(5\right) = a {\left(5\right)}^{3} + b {\left(5\right)}^{2} + c \left(5\right) - 4 = - 31.5$

$125 a + 25 b + 5 c = - 27.5$

$25 a + 5 b + c = - 5.5$

$50 a + 10 b + 2 c = - 11$

Write the 3 equations as an augmented matrix:
[ (-8,4, -2, |, 4), (8,4, 2, |, 4), (50,10,2,|,-11) ]
[ (-8,4, -2, |, 4), (0,8, 0, |, 8), (50,10,2,|,-11) ]
[ (-8,4, -2, |, 4), (0,1, 0, |, 1), (50,10,2,|,-11) ]
[ (-8,0, -2, |, 0), (0,1, 0, |, 1), (50,0,2,|,-21) ]
[ (-8,0, -2, |, 0), (0,1, 0, |, 1), (42,0,0,|,-21) ]
[ (-8,0, -2, |, 0), (0,1, 0, |, 1), (1,0,0,|,-0.5) ]
[ (0,0, -2, |, -4), (0,1, 0, |, 1), (1,0,0,|,-0.5) ]
[ (0,0, 1, |, 2), (0,1, 0, |, 1), (1,0,0,|,-0.5) ]
$a = - \frac{1}{2} , b = 1 , \mathmr{and} c = 2$

The equation is:

$y = - \frac{1}{2} {x}^{3} + {x}^{2} + 2 x - 4$