Find the real zeros of the function

Question

f(x)=6x^3-49x^2+20x-2?  

Answer 1

The function f (x ) = 6x ³ - 49x ² + 20x  - 2

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.

 6x ³ - 49x ² + 20x  - 2 = 0

If p/q  is a rational zero, then p  is a factor of 2 and q  is a factor of 6.

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

The possible values for q  are ± 1, ± 2, ± 3 and ± 6.

So, p/q  = ± 1,  ± 2, ± 1/2, ± 1/3, ± 2/3, ± 1/6.

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

p/q	6	-49	20	-2
1	6	-43	-23	-25
-1	6	-55	75	-77
2	6	-37	-54	-110
-2	6	-61	142	-286
1/2	6	-45	-5/2	-7
-1/2	6	-52	46	-25
1/3	6	-47	13/3	-5/9
1/6	6	-48	12	0

 

Answer 2

Contd...

Since f (1/6) = 0, x = 1/6 is a zero. The depressed polynomial is   6x² - 48x + 12 = 0

x² - 8x + 2 = 0

Since the depressed polynomial of this zero, x² - 8x + 2 , is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

x² - 8x + 2  = 0

Compare it to

Roots are

Real zeros are 1/6 , 7.74 and 0.26.