Find all rational zeros of f. Then (if necessary) use the depressed equation to

Question

Find all rational zeros of f. Then (if necessary) use the depressed equation to find all roots of the equation f(x) = 0. f(x) = 13x^4+ 12x^3- 144x^2- 132x+ 11 Select the correct choice below and fill in the answer box within your choice, if necessary. The set of all zeros of the given function is{ }. (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) There are no real zeros.

Explanation / Answer

f(x) = 13x^4 + 12x^3 -144x^2 -132x + 11

possible rational zeros of the function are

+- { 1 , 11 } / { 1 , 13}

+- 1 , +- 1/13 , +- 11 , +- 11/13

by plugging x = -1 , we find that one of the zero is x=-1

dividing polynomial by x+1 we get

13x^4 + 12x^3 -144x^2 -132x + 11 / (x+ 1) = 13x^3 -x^2 - 143 x + 11

another zero is x = 1/13

again dividing 13x^3 -x^2 - 143 x + 11 by (x-1/13) we get

x^2 - 11 = 0

x = +- sqrt 11

4 zeros are { -1 , 1/13 , sqrt 11, -sqrt 11 }

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