Question

According to the Rational Root Theorem, which could be a factor of the polynomial f(x) = 3x3 – 5x2 – 12x + 20?

Answer 1

a)X1=-2

b)X2=2

c)X3=5/3

Answer 2

factor of the polynomial
`f(x)=3x^(3)-5x^(2)-12x+20` is

`(x-(5)/(3))(x+2)(x-2)`

Explanation:

Rational Root Theorem: It tells us which roots we may find exactly (the rational ones) and which roots we may only approximate (the irrational ones).

`P(x) = a_n x^(n)+a_(n-1)x^(n-1) + ... + a_2 x^(2)+ a_1 x + a_0`

has any rational roots, then they must be of the form:

`\pm(factor of a_0)/(factor of a_1)`

In provided polynomial
`f(x)=3x^(3)-5x^(2)-12x+20`

Here,
`a_0=20\; \text{and}\; a_n =3`

The number 20 has factors:
`\pm1,\pm2,\pm4,\pm5,\pm10,\pm20`.

These are possible value for p

The number 3 has factors:
`\pm1,\pm3`. these are possible value for q

Find all possible value of
`(p)/(q)`

`\mathrm{The\:following\:rational\:numbers\:are\:candidate\:roots:}\quad \pm (1,\:2,\:4,\:5,\:10,\:20)/(1,\:3)`

`\mathrm{Validate\:the\:roots\:by\:plugging\:them\:into}\:3x^3-5x^2-12x+20=0:\quad x=(5)/(3),\:x=-2,\:x=2`

Hence, factor of the polynomial
`f(x)=3x^(3)-5x^(2)-12x+20` is

`(x-(5)/(3))(x+2)(x-2)`