Question

Form a polynomial whose zeros and degree are given. Multiply out to form a polynomial.
Zeros: 0, -5, 4; degree: 3

Answer 1

Welcome to Open Study!

For a polynomial, each root (or "zero") c corresponds to a factor, x-c. E.g., if I want a polynomial with roots 1 and -1, I can "cook that up" by multiplying the factors associated with each root:

\(\Large (x-1)(x-(-1))=(x-1)(x+1)=x^2-1\)

So \(\Large x^2-1\) has zeros 1, and -1, because I "cooked it up" that way. :)

Now a root of 0 just means a factor of (x-0), or just x.

So if I wanted roots of 1, -1, and 0, I need:
\(\Large x(x-1)(x+1)=x(x^2-1)=x^3-x\)

Now try that with your given roots. :)

Answer 2

Just a note to add: if you were asked for the polynomial to have a higher degree with the same roots - say, degree 5 with 3 roots - you would just make some of the roots "repeated" roots. E.g. for my example above, if I wanted degree 5 I could use:

\(\Large x^3(x-1)(x+1)\)
or \(\Large x^2(x-1)^2(x+1)\)
or \(\Large x(x-1)(x+1)^3\)

.... you get the idea, just so that the total number of factors is = the required degree.