Question

Find a degree 3 polynomial that has zeros -2,3 and 6 and in which the coefficient of x^2 is -14. The polynomial is: _____

Answer 1

Given:

The zeros of degree 3 polynomial are -2, 3 , 6.

The coefficient of x² is -14.

Let the degree 3 polynomial be,

$\begin{gathered} p(x)=(x-x_1)(x-x_2)(x-x_3) \\ =(x-(-2))(x-3)(x-6) \\ =\mleft(x+2\mright)\mleft(x-3\mright)\mleft(x-6\mright) \\ =\mleft(x^2-x-6\mright)\mleft(x-6\mright) \\ =x^3-x^2-6x-6x^2+6x+36 \\ =x^3-7x^2+36 \end{gathered}$

But given that, coefficient of x² is -14 so, multiply the above polynomial by 2.

$\begin{gathered} p(x)=x^3-7x^2+36 \\ 2p(x)=2(x^3-7x^2+36) \\ =2x^3-14x^2+72 \end{gathered}$

Answer: The polynomial is,

$p(x)=2x^3-14x^2+72$