Question

Consider the graph of the sixth-degree polynomial function f.

Answer 1

Answer:

b = 1, c = -1 and d = 4

Step-by-step explanation:

To solve this question the rule of multiplicity of a polynomial is to be followed.

If the multiplicity of a polynomial is even at a point, graph of the polynomial will touch the x-axis.

If the multiplicity of the polynomial is odd, graph will cross the x-axis at that point.

From the graph of function 'f',

f(x) = (x - b)(x - c)²(x - d)³

Since, graph of the function 'f' crosses x-axis at x = 1 and x = 4, multiplicity will be odd and touches the x-axis at x = -1 multiplicity will be even.

So the function will be,

f(x) = (x - 1)[x - (-1)]²(x - 4)³

Therefore, b = 1, c = -1 and d = 4 will be the answer.

Answer 2

Answer:

f(x)=(x-1)(x+1)^2(x-4)^3

Step-by-step explanation:

The zeros of the function are x=-1, x=1, and x=4, so its factors are x+1, x-1, and x-4.

The graph touches, but doesn’t cross, the x-axis at x=-1. So the factor x+1 is a repeated factor with an even multiplicity.

The graph crosses the x-axis at x=4, but with a little curve before and after it meets this point. So the factor x+4 is a repeated factor with odd multiplicity.

The graph crosses the x-axis at x=1 with no curve before or after. So the factor x-1 is not a repeated factor.

Therefore, function f is defined by f(x)=(x-1)(x+1)^2(x-4)^3.