Question

Find all rational zeros of the polynomial, and write the polynomial in factored form?

P(x)= 2x^3-3x^2-2x+3

Answer 1

2 x ^ 3 - 3 x ^ 2 - 2 x + 3 = 0

( 2 x ^ 3 - 2 x ) - 3 x ^ 2 + 3 = 0

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Remark :

- 3 x ^ 2 + 3 = - ( 3 x ^ 2 - 3 )

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( 2 x ^ 3 - 2 x ) - ( 3 x ^ 2 - 3 ) = 0

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

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

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Remark :

( x ^ 2 - 1 ) = ( x + 1 ) ( x - 1 )

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

2 x ^ 3 - 3 x ^ 2 - 2 x + 3 = ( x + 1 ) ( x - 1 ) ( 2 x - 3 ) = 0


x + 1 = 0

x = - 1


x - 1 = 0

x = 1


2 x - 3 = 0

2 x = 3

x = 3 / 2


The solutions are :

x = - 1

x = 1

and

x = 3 / 2