Find a polynomial function of degree 3 that has zeros of -5, 3, and 4.

Answer 1

Background Information:

1. A polynomial function is a function which has at least one algebraic term.
2. When you say that a polynomial function is of the third degree, it means that the largest exponent in the function is 3. For example x³ is a polynomial function (as it has at least one algebraic term) and it is to the third degree because the largest exponent is 3.
3. The zeros of a function are the roots of the function. To find the roots of a function you set the function equal to zero and you solve for which values for your variables make the function true. For example if you have the function (x-5)(x+4)=0, your roots are 5 and -4, because for both of these values your function is equal to zero.

Solving the problem

The problem is asking you to find a polynomial function of the third degree that has the zeros (or roots) -5, 3, and 4. Using our background information, we can easily solve this problem.

1: Set your variable equal to the roots:

For this problem, let us use x as our variable.

x = -5, 3, 4

2: Show how these values are our roots by creating functions where, using these values, the function is equal to zero.

x = -5 => x+5=0

x = 3 => x-3=0

x = 4 => x-4=0

3: Since we know that the function we are looking for is a polynomial of which all of these values allow the function to be equal to zero, we combine our functions to create the polynomial of the third degree. But how should we combine it? Do we add, subtract, multiply, or divide? Well if we add or subtract the functions from each other it will not be to the third degree, as the largest exponent will still be 1 (making it a 1st degree function) and the zeros will no longer be true for the function. If we divide the functions it will also not be to the third degree, and the zeros will no longer be true for the function (these are only 2 of the many issue that can come up if you attempted to divide). So that leaves us with multiplication, which makes sense since multiplying the functions (which are equal to zero) will give us a third degree function that is equal to zero. So let us do that, let us multiply the functions.

(x+5)(x-3)(x-4) = 0

The function above is the answer to the problem. It is a polynomial function of the third degree with the zeros = -5, 3, and 4. To prove it is to the third degree, below is the expanded function:

x³ − 5x² − 16x + 80 = 0

As you can see, the largest exponent is 3.