The rational root theorem is used to determine the possible roots of a polynomial
- Using the rational root theorem, the potential roots are: [tex]\pm (1,2,4,8, \frac 12, \frac 13,\frac 23,\frac 43,\frac 83, \frac 14, \frac 15,\frac 25,\frac 45,\frac85, \frac 16,\frac{1}{10} , \frac{1}{12}, \frac{1}{15},\frac{2}{15},\frac{4}{15},\frac{8}{15}, \frac{1}{20}, \frac{1}{30},\frac{1}{60})[/tex]
- The factors of the polynomial are [tex](2x^2 -1), (6x - 1)[/tex] and [tex](5x + 8)[/tex]
How to determine the roots
The polynomial function is given as:
[tex]f(x) = 60x^4 + 86x^3 - 46x^2 - 43x + 8[/tex]
Using the rational root theorem, the potential roots are:
[tex]Roots = \pm \frac{Factors\ of\ 8}{Factors\ of\ 60}[/tex]
List the factors of 8 and 60
[tex]Roots = \pm \frac{1,2,4,8}{1,2,3,4,5,6,10, 12, 15, 20, 30, 60}[/tex]
Expand the above expressions
[tex]Roots = \pm (\frac{1,2,4,8}{1}, \frac{1,2,4,8}{2}, \frac{1,2,4,8}{3}, \frac{1,2,4,8}{4}, \frac{1,2,4,8}{5}, \frac{1,2,4,8}{6},\frac{1,2,4,8}{10}, \frac{1,2,4,8}{12}, \frac{1,2,4,8}{15}, \frac{1,2,4,8}{20}, \frac{1,2,4,8}{30}, \frac{1,2,4,8}{60})[/tex]
Simplify each quotient
[tex]Roots = \pm ((1,2,4,8), (\frac 12,1,2,4), (\frac 13,\frac 23,\frac 43,\frac 83), (\frac 14,\frac 12,1,2), \\(\frac 15,\frac 25,\frac 45,\frac85),( \frac 16, \frac 13 , \frac 23 , \frac 43),(\frac{1}{10} ,\frac{1}{5},\frac{2}{5},\frac{4}{5}), (\frac{1}{12},\frac 16,\frac 13,\frac 23), \\(\frac{1}{15},\frac{2}{15},\frac{4}{15},\frac{8}{15}), (\frac{1}{20},\frac{1}{10},\frac15,\frac{2}{4}), \frac{1}{30},\frac{1}{15},\frac{2}{15},\frac{4}{15}), (\frac{1}{60},\frac{1}{30},\frac{1}{15},\frac{2}{15})[/tex]
Remove repetition from the above list
[tex]Roots = \pm (1,2,4,8, \frac 12, \frac 13,\frac 23,\frac 43,\frac 83, \frac 14, \frac 15,\frac 25,\frac 45,\frac85, \frac 16,\frac{1}{10} , \frac{1}{12}, \frac{1}{15},\frac{2}{15},\frac{4}{15},\frac{8}{15}, \frac{1}{20}, \frac{1}{30},\frac{1}{60})[/tex]
How to determine the factors
To determine the factors, we have:
[tex]f(x) = 60x^4 + 86x^3 - 46x^2 - 43x + 8[/tex]
Expand the polynomial
[tex]f(x) = 60x^4 + 86x^3 -30x^2 -16x^2 - 43x + 8[/tex]
Rearrange the terms of the polynomial
[tex]f(x) = 60x^4 + 86x^3 -16x^2 -30x^2 - 43x + 8[/tex]
Group into two
[tex]f(x) = [60x^4 + 86x^3 -16x^2] -[30x^2 + 43x - 8][/tex]
Factor each group
[tex]f(x) = 2x^2(30x^2 + 43x -8) -1(30x^2 + 43x - 8)[/tex]
Factor out 30x^2 + 43x - 8
[tex]f(x) = (2x^2 -1)(30x^2 + 43x - 8)[/tex]
Expand 30x^2 + 43x - 8
[tex]f(x) = (2x^2 -1)(30x^2 + 48x - 5x - 8)[/tex]
Factorize
[tex]f(x) = (2x^2 -1)(6x(5x + 8) -1(5x + 8))[/tex]
Factor out 5x + 8
[tex]f(x) = (2x^2 -1)(6x - 1)(5x + 8)[/tex]
Hence, the factors of the polynomial are [tex](2x^2 -1), (6x - 1)[/tex] and [tex](5x + 8)[/tex]
Read more about rational root theorem at:
https://brainly.com/question/1353541
