Answer:
Step-by-step explanation:
Those zeros are solutions of the polynomial which are found by factoring. The solutions can then be written as factors, which can then be multiplied together to get back the polynomial. Because you have 5 solutions (zeros, roots) this is a 5th degree polynomial. Since we have
x = 0, multiplicity 3, those are where we will start because those are the easy ones. The factor of x = 0 is either x - 0 = 0 or x + 0 = 0, the factor being (x - 0).
We will multiply the first 2 factors together, then multiply in the third:
(x - 0)(x - 0) = x² - 0x - 0x - 0 which is, of course, just x².
Now multiplying in the third (x - 0):
x²(x - 0) = x³ - 0x² which is just x³.
We'll leave that for a minute and go to the next zeros, which are conjugates of one another and will always follow the rule that if there is x + a radical (or an imaginary number), there will ALWAYS be x - a radical (or imaginary number). Those are, as zeros, x = √3 and x = - √3. As factors, they are
(x + √3) and (x - √3). We will multiply those together:
x² + x√3 - x√3 - √9
The middle terms cancel each other out, and the square root of 9 is 3, so we simplify this down to:
x² - 3.
Now we can finish the multiplying:
x³(x² - 3) = x⁵ - 3x³
