Question

Obtain a graph of the function for the various values of a, b, and c in the following table. Conjecture a relation between the degree of a polynomial function and the number of turning points after completing the table. In the table, a can be 1, 2, or 3; b can be 1, 2, or 3; and c can be 1, 2, 3, or 4.

Short answer

From the table, we see that the turning point is always less than the degree.

Also, if the number of all zero point is odd then the number of turning points is equal to 2, and for each zero point that is of even multiplicity the number of turning points increases by one.

Step-by-step solution

Step 1. Given Information

We need to Obtain a graph of the function for the various values of a, b, and c in the following table. Conjecture a relation between the degree of a polynomial function and the number of turning points after completing the table. In the table, a can be 1, 2, or 3; b can be 1, 2, or 3; and c can be 1, 2, 3, or 4.

Step 2. For a=1,b=1,c=1

$f\left(x\right)=\left(x+2\right)x\left(x-2\right)$

The degree of the polynomial is degree 3 and the turning point is 2.

Step 3. For a=1,b=1,c=2

$f\left(x\right)=\left(x+2\right)x{\left(x-2\right)}^2$

The degree of the polynomial is degree 4 and the turning point is 3.

Step 4. For a=1,b=1,c=3

$f\left(x\right)=\left(x+2\right)x{\left(x-2\right)}^3$

The degree of the polynomial is degree 5 and the turning point is 2.

Step 5. Forming more polynomials

Continue in this manner, we can construct the table