SOLUTION
The given function is:
[tex]f(x)=3x^4+2x^3-22x^2-14x+7[/tex]
Recall the rule of rational zero theorem
[tex]\frac{p}{q}=\frac{a\text{ factor of the constant term}}{a\text{ factor of the leading cooficient}}[/tex]
From the given function
The contant term is 7
Factors of the constant term are
[tex]\pm1,\pm7[/tex]
The leading coefficient is 3
Factors of the leading coefficient are
[tex]\pm1,\pm3[/tex]
Using the the rational zero theorem
The posible zeros are
[tex]\frac{\pm1}{\pm3},\frac{\pm1}{\pm1},\frac{\pm7}{\pm3},\frac{\pm7}{\pm1}[/tex]
This gives
[tex]\pm\frac{1}{3},\pm1,\pm\frac{7}{3},\pm7[/tex]
Substitute the possible zeros into the equation to get the actual zeros
The roots are
[tex]\frac{1}{3},-1[/tex]
Factorizing the given function gives
[tex]f(x)=(x+1)(3x-1)(x^2-7)[/tex]
