According to the Rational Root Theorem, which could be a factor of the polynomial f(x) = 3x3 – 5x2 – 12x + 20?
Answer:
factor of the polynomial \(f(x)=3x^{3}-5x^{2}-12x+20\) is
\((x-\frac{5}{3})(x+2)(x-2)\)
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).
\(P(x) = a_n x^{n}+a_{n-1}x^{n-1} + ... + a_2 x^{2}+ a_1 x + a_0\)
has any rational roots, then they must be of the form:
\(\pm\frac{factor of a_0}{factor of a_1}\)
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 \(\frac{p}{q}\)
\(\mathrm{The\:following\:rational\:numbers\:are\:candidate\:roots:}\quad \pm \frac{1,\:2,\:4,\:5,\:10,\:20}{1,\:3}\)
\(\mathrm{Validate\:the\:roots\:by\:plugging\:them\:into}\:3x^3-5x^2-12x+20=0:\quad x=\frac{5}{3},\:x=-2,\:x=2\)
Hence, factor of the polynomial \(f(x)=3x^{3}-5x^{2}-12x+20\) is
\((x-\frac{5}{3})(x+2)(x-2)\)