Question

What are all the rational roots of the polynomial f(x) = 20x4 + x3 + 8x2 + x – 12?

Answer 1

Answer:

-0.8 and 0.75.

Step-by-step explanation:

20x^4 + x^3 + 8x^2 + x – 12 = 0

by the rational roots theorem:-

The leading coefficient is 20 with factors +/-  1,2,4,5,10,20

and the constant is -12 with factors  +/- 1,2,3,4,6,12

so there are many possible roots like -1, 2, 3/4 3/5 , -4/5 .....

Drawing a graph would be the quickest way to do this,  either manually, using graphical software or a graphical calculator.

I used the latter and got 2 rational roots -0.8 and 0.75.

Answer 2

Answer:

$x=\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{1}{10},\\\\\pm\frac{1}{5},\pm\frac{3}{5}\pm\frac{3}{10},\pm\frac{2}{5},\pm\frac{6}{5},\pm\frac{1}{20},\pm\frac{3}{20},\pm\frac{4}{5},\pm\frac{12}{5}$                                                                

Step-by-step explanation:

Given : Polynomial $f(x)=20x^4+x^3+8x^2+x-12$

To find : What are all the rational roots of the polynomial?

Solution :

Polynomial $f(x)=20x^4+x^3+8x^2+x-12$

The Rational Root Theorem states that the all possible roots of a polynomial are in the form of a rational number $\frac{p}{q}$ form

Where, p is a factor of constant term

and q is the factor of coefficient of leading term.

On comparing with given polynomial,

The constant term is p=-12

All possible factor of 12 are $\pm1,\pm2,\pm3,\pm4,\pm6,\pm12$

Leading coefficient is q=20

All possible factor of 20 are $\pm1,\pm2,\pm4,\pm5,\pm10,\pm20$

Now, all possible rational roots of the given polynomial are,

$x=\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{1}{10},\\\\\pm\frac{1}{5},\pm\frac{3}{5}\pm\frac{3}{10},\pm\frac{2}{5},\pm\frac{6}{5},\pm\frac{1}{20},\pm\frac{3}{20},\pm\frac{4}{5},\pm\frac{12}{5}$