Find all the zeros of the following function y=x^4+6x^3+6x^2+6x+5

Then the four zeros of the given polynomial are {-1, -5, -1, +1}

Step-by-step explanation:

First, make an educated guess regarding the first zero of this polynomial.  Since the constant term is 5, likely zeros are -1, 1, -5 and 5.  Let's check out the possible zero -1, using synthetic div.:

-1   /   1   6   6   6   5

             -1   -5  -1   -5

        1     5    1    5   0    These are the coefficients of the quotient.

Because the remainder is zero, we know that -1 is a zero of the given polynomial.

Next, let's determine whether -5 is a zereo of the above quotient:

-5   /  1    5    1    5

              -5   0  -5

        1      0    1    0  Because the remainder is zero, -5 is a zero.

Here the quotient is 1x^2 - 1^2, which factors into (x+1)(x-1)

Setting this result = to 0, we get x = -1 and x = + 1.

Then the four zeros of the given polynomial are {-1, -5, -1, +1}

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step-by-step explanation: