Find a polynomial with the given zeros. $0,0,0 , \sqrt { 6 } , - \sqrt { 6 }$
A) $x ^ { 5 } - x ^ { 3 } + 6 x$
B) $\quad x ^ { 5 } - 3 x ^ { 3 }$
C) $\quad x ^ { 2 } + 6 x + 3$
D) $x ^ { 5 } - 6 x ^ { 3 }$
E) none of these

Question

Quizplus · Accepted Answer

A polynomial with a root of 0 has a factor of x. Since 0 occurs three times, we have a factor of x^3. The other two roots are square roots of 6, so we have factors of (x-sqrt(6)) and (x+sqrt(6)). Therefore, our polynomial is:
$$f(x) = kx^3(x-\sqrt{6})(x+\sqrt{6})$$
Simplifying gives us:
$$f(x) = kx^5 - 6kx^3$$
The only answer choice that matches this form is D.

Find a Polynomial with the Given Zeros 0,0,0,6,−60,0,0 , \sqrt { 6 } , - \sqrt { 6 }0,0,0,6​,−6​

Find a Polynomial with the Given Zeros $0,0,0 , \sqrt { 6 } , - \sqrt { 6 }$