Question

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

Answer 1

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

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}