A polynomial f(x) has a factor x-k if and only if f(k)=0.
This theorem tells us that if k is a zero of a polynomial function, then x-k is one of its factors. We can use this fact to write a cubic equation with repeated zeros. For example, suppose we want the related function to have a zero at x=0 which repeats 3 times. The factor (x-0) will appear three times in f(x).
f(x) = (x-0)(x-0)(x-0) ⇔ f(x) = x^3
Therefore, the related cubic equation x^3=0 has a repeated solution, as required. We could use a similar procedure and chose the repeating roots to obtain a quartic equation this time. For example, x= 0, x= 0, x= 1, and x= -1.
(x- 0)(x- 0)(x-( -1))(x- 1) = 0
⇕
(x)(x)(x+1)(x-1) = 0
We can expand the product to write the equation in standard form.
We have found the quartic equation x^4-x^2 = 0, which has repeated solutions, as required. Notice that these are just some example equations. There are infinitely many equations satisfying the exercise requirements.
