Find all rational zeros of h(x)= 3x^3+7x^2+8x+2.

1 Answer

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 h (x ) = 3x³+ 7x²+ 8x + 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, ± 1/3, ± 2 and ± 2/3.

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

p /q	3	7	8	2
1	3	10	18	20
-1	3	4	4	-2
1/3	3	8	32/3	34
-1/3	3	6	6	0

Since h (-1/3) = 0, x = -1/3 is a zero. The depressed polynomial is 3x²+ 6x + 6 = 0

The above equation can be written as x² + 2x + 2 = 0

Factoring of h (x ) = 3x³+ 7x²+ 8x + 2 = (x + 1/3)(x² + 2x + 2)

Since the depressed polynomial of this zero, x² + 2x + 2, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

Roots are