Using the rational zeros theorem to find all zeros of a polynomial

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The function below has at least one rational zero. Use this fact to find all zeros of the function.

h(x)= 6x^3 - 43x^2 + 6x + 7

If there is more than one zero separate them with commas. Write exact values, not decimal approximations.

Oct 17th, 2015

Hello!

The possible rational zeros are

+-1, +-1/2, +-1/3, +-1/6, +-7, +-7/2, +-7/3, +-7/6.

Test x=7: h(7)=6*7^3 - 43*7^2 + 6*7 + 7 = (42-43)*7^2 + 49 = -49 + 49 = 0.
So 7 is a root of h.

Now divide h by (x-7):

6x^3 - 43x^2 + 6x + 7 = 6x^2(x-7) + 42x^2 - 43x^2 + 6x + 7 = 
6x^2(x-7) - x^2 + 6x + 7 = 6x^2(x-7) - x(x-7) - 7x + 6x + 7 = 
6x^2(x-7) - x(x-7) - x + 7 = 6x^2(x-7) - x(x-7) - (x - 7) = 
(x-7)(6x^2-x-1).

Now solve the quadratic equation 6x^2-x-1=0:
x1,2 = (1+-sqrt(25))/12 = 1/2 and -1/3.

The answer: 7, 1/2, -1/3.

Please ask if something is unclear.
Oct 17th, 2015

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