David W. answered • 07/15/16

Experienced Prof

A polynomial may be written as (x-a)(x-b)(x-c)...(x-n)=0. Each of the constants (a, b, c, ... n) represent zeros. The product of the expression on the left side is zero (to match the right side of the equation) because of the Zero Product Property (anything multiplied by zero results in a zero). Polynomials written this way are in what is called

**Factored Form**.So, if the zeros occur at 1/2, 2/3, and 2, the polynomial is:

(x-1/2)(x-2/3)(x-2)=0

(2x-1)(3x-2)(x-2)=0 [multiply factors (and right side) by 2 and by 3 rid of fractions]

To write the polynomial in

**Standard From**(with**integer coefficients**), simply multiply these terms [note: use distributive principle { a(b+c)=ab+ac } and F-O-I-L ]: ( (2x-1)(3x-2) )(x-2) = 0

( (2x-1)((3x-2) )x - 2( (2x-1)(3x-2) ) = 0 [distribute]

(6x

^{2}- 4x -3x + 2)x - 2(6x^{2}- 4x - 3x + 2) = 0 [F-O-I-L] 6x

^{3}- 7x^{2}+ 2x - 12x^{2}+ 14x - 4 = 0 [distribute] 6x

^{3}-19x^{2}+ 16x - 4 = 0 [collect like terms] [note that the

**coefficients are all integers**]