Question

Algebra II Question. Please explain your answer.

Answer 1

0 is obviously a solution if you substitute 0 for x 

Test 3 

3 * 3^4 - 3^3 - 27 * 3^2 + 9 * 3 = 0 that 3 is a root. 

continue testing the remainder.

Answer 2

First, we could try to factorize the polynomial. We could group the first and third terms, and the second and fourth term:


$\displaystyle{ (3x^4-27x^2)-(x^3-9x)=3x^2(x^2-9)-x(x^2-9)$

Now, x(x^2-9) is a common factor:

$\displaystyle{ x(3x-1)(x^2-9)$.

By the difference of squares formula, our final factorization is :

x(3x-1)(x-3)(x+3).

Here we can see clearly that the roots of f(x) are 0, 1/3, 3, and -3.



Remark: we grouped the terms in pairs as we did, noticing that the ratio of the exponents, and coefficient in each pair was the same. 


Answer: 0, 1/3, 3, and -3.