Find a formula for the polynomial of least degree through th

Find a formula for the polynomial of least degree through the points shown in the graph.

Solution

There are three turning points , Hence the Degree of polynomial is 4.

Graph has two real roots

At x = 1 , graph is bouncing back ==> multiplicity of root 1 is 2

At x = -4 , graph is bouncing back ==> multiplicity of root -4 is 2

==> Polynomial with degree 4 and roots 1 , -4 with multiplicities 2 is

p(x) = (x -1)² (x +4)²

==> p(x) = (x² -2x +1) (x² + 8x + 16)

==> p(x) = x⁴ + 8x³ + 16x² -2x³ -16x² -32x + x² + 8x + 16

==> p(x) = x⁴ + 6x³ + x² -24x + 16