according to the rational root theorem, what are all the potential rational roots of (f(x)=9x^{4}-2x^{2}-3x…

according to the rational root theorem, what are all the potential rational roots of (f(x)=9x^{4}-2x^{2}-3x + 4)?\n(pm\frac{1}{9},pm\frac{2}{9},pm\frac{1}{3},pm\frac{4}{9},pm\frac{2}{3},pm1,pm\frac{4}{3},pm2,pm4)\n(pm\frac{1}{4},pm\frac{1}{2},pm\frac{3}{4},pm1,pm\frac{3}{2},pm\frac{9}{4},pm3,pm\frac{9}{2},pm9)\n(\frac{1}{9},\frac{2}{9},\frac{1}{3},\frac{4}{9},\frac{2}{3},1,\frac{4}{3},2,4)\n(\frac{1}{4},\frac{1}{2},\frac{3}{4},1,\frac{3}{2},\frac{9}{4},3,\frac{9}{2},9)

according to the rational root theorem, what are all the potential rational roots of (f(x)=9x^{4}-2x^{2}-3x + 4)?\n(pm\frac{1}{9},pm\frac{2}{9},pm\frac{1}{3},pm\frac{4}{9},pm\frac{2}{3},pm1,pm\frac{4}{3},pm2,pm4)\n(pm\frac{1}{4},pm\frac{1}{2},pm\frac{3}{4},pm1,pm\frac{3}{2},pm\frac{9}{4},pm3,pm\frac{9}{2},pm9)\n(\frac{1}{9},\frac{2}{9},\frac{1}{3},\frac{4}{9},\frac{2}{3},1,\frac{4}{3},2,4)\n(\frac{1}{4},\frac{1}{2},\frac{3}{4},1,\frac{3}{2},\frac{9}{4},3,\frac{9}{2},9)

Answer

Explanation:

Step1: Recall Rational Root Theorem

The Rational Root Theorem states that if a polynomial (f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0) has integer coefficients, then the possible rational roots are of the form (\pm\frac{p}{q}), where (p) is a factor of the constant term (a_0) and (q) is a factor of the leading - coefficient (a_n).

Step2: Identify (a_0) and (a_n)

For the polynomial (f(x)=9x^4 - 2x^2-3x + 4), the leading - coefficient (a_n = 9) and the constant term (a_0 = 4).

Step3: Find factors of (a_0) and (a_n)

The factors of (a_0 = 4) are (p=\pm1,\pm2,\pm4), and the factors of (a_n = 9) are (q=\pm1,\pm3,\pm9).

Step4: Calculate possible rational roots

The possible rational roots (\frac{p}{q}) are (\pm\frac{1}{9},\pm\frac{2}{9},\pm\frac{1}{3},\pm\frac{4}{9},\pm\frac{2}{3},\pm1,\pm\frac{4}{3},\pm2,\pm4).

Answer:

(\pm\frac{1}{9},\pm\frac{2}{9},\pm\frac{1}{3},\pm\frac{4}{9},\pm\frac{2}{3},\pm1,\pm\frac{4}{3},\pm2,\pm4)