Question

Which Of The Following Is A Root Of The Polynomial g(x)=2x^3-19x^2+64x-60.

Answer 1

g(x ) = 2x ³- 19x ² + 64x - 60

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.

 g (x ) = 2x ³- 19x ² + 64x - 60

If p /q  is a rational zero, then p  is a factor of 60 and q  is a factor of 2.

The possible values of p  are   ± 1, ± 2.  ± 3, ± 4,± 5,± 6,± 10,± 12, ± 15 and ± 30 .

The possible values for q are ± 1,± 2.

So, p /q  = ± 1, ± 2, ± 3/2, ± 5/2.

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

p / q	1	1	-1	15
1	2	-17	47	-13
-1	2	-21	85	-145
2	2	-15	34	-8
-2	2	-23	100	-260
3/2	2	-16	40	0

Since g (3/2)   =   0,  x = 3/2 is a zero. The depressed polynomial is   2x ²- 16x + 40 = 0

Since the depressed polynomial of this zero, 2x ²- 16x + 40, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

The polynomial  function roots are