Find a polynomial function that has the given zeros. (There are many correct answers.) (0,1,3

Question

f(x) =

Answer 1

If you have A*B*C = 0, then either A=0 or B = 0 or C = 0.

If I have a function y = f(x) that does not have a zero at x = 1, then I can multiply f(x) by (x-1) and the resulting function will have a zero at x = 1:

f(1) * (x-1) = f(1) * 0 = 0

We need a function that also has a zero at x = 3. So multiply our function by (x-3) also:

f(x)*(x-1)*(x-3) = f(3)*(2)*(0) = 0.

Now we have a function with a zero at 1 and a zero at 3. If you then multiply this function by (x-0), then you'll have a zero at x = 0 as well:

f(x)(x-1)(x-3)(x-0) = f(0)(0-1)(0-3)(0-0) = 0.

You can start with any function f, and this will work. You can even start with the trivial function f(x) = 1.

So, this:

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

is the simplest solution

Answer 2

A polynomial function's zeros are also called its roots or solutions. They are easily found from setting the fully factored polynomial equal to zero. To create a polynomial from given zeros, simply reverse the process.

Here our roots are 0,1, 3 while there are many polynomials with those roots, perhaps the simplest is

(x-0)(x-1)(x-3)=0 in factored form

this is the same as

x(x-1)(x-3) =0

Now we expand (reverse of factoring)

x(x^2 -4x +3)=0

x^3 - 4x^2 +3x = 0

to check if this polynomial has the given zeros, try each value as x and verify that it equals 0