Question

Find all the zeroes of polynomial 2x4

Answer 1

9514 1404 393

Answer:

  x ∈ {-1/2, 1, 2-√3, 2+√3}

Step-by-step explanation:

The given zeros mean that one of the quadratic factors of the given polynomial is ...

  (x -2)^ -3 = x^2 -4x +1

When that is factored out (see first attachment), the remaining quadratic is ...

  2x^2 -x -1

This can be factored as ...

  = (2x +1)(x -1)

which has roots that make these factors zero: x = -1/2, x = 1.

So, all of the zeros of the given polynomial are ...

  -1/2, 1, 2-√3, 2+√3 . . . all zeros

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A graphing calculator can often point to the zeros of the function quite nicely.

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Additional comments

When p is a zero of a polynomial, (x-p) is a factor of it. The given zeros mean that factors are (x-2-√3) and (x-2+√3). The product of these factors is the difference of the squares (x-2)^2 and (√3)^2, so is (x -2)^2 -3.

Using the pattern for the square of a binomial, we see this is ...

  (x-2)^2 = x^2 -2·2x +2^2 = x^2 -4x +4

The product of the given factor is then 3 subtracted from this square. The given zeros mean there is a quadratic factor of ...

  (x-2-√3)(x-2+√3) = (x-2)^2 -3 = x^2 -4x +4 -3 = x^2 -4x +1