Respuesta :
Answer:
factor of the polynomial [tex]f(x)=3x^{3}-5x^{2}-12x+20[/tex] is
[tex](x-\frac{5}{3})(x+2)(x-2)[/tex]
Step-by-step 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).
[tex]P(x) = a_n x^{n}+a_{n-1}x^{n-1} + ... + a_2 x^{2}+ a_1 x + a_0[/tex]
has any rational roots, then they must be of the form:
[tex]\pm\frac{factor of a_0}{factor of a_1}[/tex]
In provided polynomial [tex]f(x)=3x^{3}-5x^{2}-12x+20[/tex]
Here, [tex]a_0=20\; \text{and}\; a_n =3[/tex]
The number 20 has factors: [tex]\pm1,\pm2,\pm4,\pm5,\pm10,\pm20[/tex].
These are possible value for p
The number 3 has factors: [tex]\pm1,\pm3[/tex]. these are possible value for q
Find all possible value of [tex]\frac{p}{q}[/tex]
[tex]\mathrm{The\:following\:rational\:numbers\:are\:candidate\:roots:}\quad \pm \frac{1,\:2,\:4,\:5,\:10,\:20}{1,\:3}[/tex]
[tex]\mathrm{Validate\:the\:roots\:by\:plugging\:them\:into}\:3x^3-5x^2-12x+20=0:\quad x=\frac{5}{3},\:x=-2,\:x=2[/tex]
Hence, factor of the polynomial [tex]f(x)=3x^{3}-5x^{2}-12x+20[/tex] is
[tex](x-\frac{5}{3})(x+2)(x-2)[/tex]
