If 'a' is a zero of the polynomial, then (x -a) will be a factor. For the given zeros, the simplest polynomial will be the product of the corresponding factors:

p(x) = (x -2)(x +2)(x -4)(x -1) . . . . . . note that x -(-2) = x +2 (factored form)

Multiplying these out gives the result in standard form.

The product of the <em>first two factors</em> is a "special product" recognizable as the difference of two squares.

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

The product of the <em>last two factors</em> can be found in the usual way. The distributive property applies.

(x -4)(x -1) = x(x -1) -4(x -1)

= x^2 -x -4x +4 = x^2 -5x +4

Then the full polynomial is the product of these partial products:

p(x) = (x^2 -4)(x^2 -5x +4)

= x^2(x^2 -5x +4) -4(x^2 -5x +4)

= x^4 -5x^3 +4x^2 -4x^2 +20x -16

p(x) = x^4 -5x^3 +20x -16 . . . . . . . . standard form

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2 years ago