# the function f(x)=3x^3-2x^2-7x-2 has at least one rational zero. use this fact to find all the zeros of g(x)

rational zeros for 3x^3-2x^2-7x-2

+1 vote

For x +1 to be a factor, you must have x = -1 as a zero. Using this information, do the synthetic division with  x = -1 as the test zero on the left:

-1  |     3        -2           -7             -2

|                 -3         5                2

3        -5         -2               0

Since the remainder is zero, then  x = -1 is indeed a zero of 3x^3-2x^2-7x-2

The polynomial can written as (x+1)(3x^2-5x-2)

Fatorize the polynomial 3x^2-5x-2

=3x^2-6x+x-2

=3x(x-2)+1(x-2)

=(x-2)(3x+1)

The polynomial can written as (x+1)(x-2)(3x+1)

factors are -1,2,-1/3

Identify Rational Zeros

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

The function f (x ) = 3x3 - 2x2- 7x - 2

If p/q is a rational zero, then p  is a factor of 2 and q  is a factor of 3.

The possible values of p  are   ± 1 and  ± 2.

The possible values for q  are ± 1 and  ± 3.

So, p/q = ± 1, ± 2, ± 1/3 and ± 1/3.

Make a table for the synthetic division and test possible  zeros.

 p/q 3 -2 -7 -2 1 3 1 -6 -8 -1 3 -5 -2 0

Since f (-1) = 0,  = -1 is a zero. The depressed polynomial is  3x2- 5x - 2 .

Since the depressed polynomial of this zero, 3x2- 5x - 2, is quadratic, use the factorization method to find the roots of the related quadratic equation 3x2- 5x - 2 = 0.

3x2- 6x + x - 2 = 0

3x (x - 2) + 1 (x - 2) = 0

(x - 2)(3x + 1) = 0

Apply zero product property.

x - 2 = 0 and 3x +1 = 0

x = 2 and 3x = -1

x = 2 and x = -1/3.

Rational zeros of f (x ) = 3x3 - 2x2- 7x - 2 are at x = - 1 , 2 and - 1/3.