4. Mrs. Collins is at the table with you and states that the fourth-degree graphs she has seen have 4 real zeros. She asks you if it is possible to create a fourth-degree polynomial with only 2 real zeros. Demonstrate how to do this and explain your steps.

Question

4.	Mrs. Collins is at the table with you and states that the fourth-degree graphs she has seen have 4 real zeros. She asks you if it is possible to create a fourth-degree polynomial with only 2 real zeros. Demonstrate how to do this and explain your steps.

agl202 · Answer

You can create a fourth degree polynomial with only two real zeros.  The other two can be complex zeros.

agl202 · Answer

Start with a quadratic function: ax^2 + bx + c. 
A quadratic function will have complex roots when the discriminant is negative.  Discriminant is b^2 - 4ac
Choose suitable values for a, b and c such that b^2 - 4ac < 0.
Then the quadratic equation will have two complex roots.

Then take another quadratic that has two real roots.
For example, (x - 1)(x - 2) is a quadratic with two real roots: 1 and 2.

Multiply the first and the second quadratic and you will have a fourth degree polynomial with two real roots and two complex roots.

Hope this helps!!!:)

anonymous · Answer

Wow thank you for your time

agl202 · Answer

Ur welcome! Glad to help u! =)

anonymous · Answer

3.	Dr. Collier summons you over to his table. He wants to demonstrate the graph of a fourth-degree polynomial function, but the batteries in his graphing calculator have run out of juice. Explain to Dr. Collier how to create a rough sketch of a graph of a fourth-degree polynomial function..

agl202 · Answer

It is possible to create a fourth-degree polynomial with ONLY two real zeros.

A a fourth-degree polynomial with have a total of 4 roots.  If only 2 roots are real then the other 2 must be complex.

How to create such a polynomial?

First find a quadratic function with two complex roots and then find another quadratic function with two real roots.  Multiply the two quadratics together and you will have a fourth-degree polynomial with 2 real roots and 2 complex roots.


ranga Best Response  Medals 28
Example: (x-1)(x-2) will have two real roots 1 and 2 because if you put either value for x, (x-1)(x-2) will be 0.
Multiply it out and you have the quadratic x^2 - 3x + 2 with two real roots

Take another quadratic, for example, x^2 + 2x + 3.  This has two complex roots.  How do I know that?  Because the discriminant (b^2 - 4ac) is negative for this function.

To get the 4th degree polynomial multiply the above two quadratics and you will end up with a fourth-degree polynomial with 2 real roots and 2 complex roots.

(x^2 - 3x + 2)(x^2 + 2x + 3)

IDK if they actually want a fourth-degree polynomial or just the steps as to how you would go about creating one.  But if they do actually want one just multiply the above two quadratics and simplify it.

Hope this helps ya =)

anonymous · Answer

Ok this is the last question so thank you again for your help.

agl202 · Answer

Anytime...