Find the real zeros of the polynomial function f(x)=x^4+6x^3-3x^2-52x-60

Question

 Find the real zeros of the polynomial function f(x)=x^4+6x^3-3x^2-52x-60

Answer 1

Try using the rational root theorem and synthetic division.

 

-2 | 1      6       -3         -52        -60

 

             -2       -8          22         60

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      1      4       -11        -30          0

 

 

(x + 2)(x³ + 4x² - 11x - 30)

 

 

Then use synthetic division on the cubic factor to find a quadratic factor.

 

3 | 1       4           -11             -30

 

              3            21              30

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      1      7            10               0

 

 

(x + 2)(x - 3)(x² + 7x + 10)

 

 

This factors completely to

 

(x + 2)(x - 3)(x + 2)(x + 5)

 

 

Set the 4 factors equal to zero and solve for x for each of them.  You already two of the zeros from doing the synthetic division.