Answer :
Answer:
Step-by-step explanation:
2 real roots and 2 complex
Following are the solution to the given polynomial equation:
Given:
Polynomial equation: [tex]\bold{n(x) = 3x^4 - 9x^3 + x^2 -3x}[/tex]
Factor: [tex]\bold{(x -3)}[/tex]
To find:
A number of roots for the polynomial n(x).
Solution:
Polynomial equation: [tex]\bold{n(x) = 3x^4 - 9x^3 + x^2 -3x}[/tex]
[tex]\to \bold{(x -3)=0}\\\\\to \bold{x =3}\\\\[/tex]
Putting the of x into the given Polynomial function:
[tex]\to \bold{n(3) = 3(3)^4 - 9(3)^3 + (3)^2 -3(3)}\\\\\to \bold{n(3) = 3\times 81 - 9\times 27 + 9 -3 \times 3}\\\\\to \bold{n(3) = 243- 243 + 9 -9}\\\\\to \bold{n(3) = 0}\\\\[/tex]
Let divide the Polynomial function by the factor:
[tex]\to \bold{\frac{3x^4 - 9x^3 + x^2 - 3x}{x-3}}\\\\\to \bold{\frac{ x(3x^3 - 9x^2 + x - 3)}{x-3}}\\\\\to \bold{\frac{ x(3x^2(x - 3)+1(x - 3))}{x-3}}\\\\\to \bold{\frac{ x(x - 3) (3x^2+1)}{x-3}}\\\\\to \bold{x(3x^2+1)}\\\\[/tex]
So, the factors are [tex]\bold{x, (x-3),\ and \ (3x^2+1)}[/tex].
Calculation of the roots:
[tex]\bold{x=0 \ \ \ \ \ \ \ \ \ \ \ \ \ x-3=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3x^2+1 =0}\\\\\bold{x=0 \ \ \ \ \ \ \ \ \ \ \ \ \ x=3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3x^2 = -1}\\\\\bold{x=0 \ \ \ \ \ \ \ \ \ \ \ \ \ x=3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 = -\frac{1}{3}}\\\\\bold{x=0 \ \ \ \ \ \ \ \ \ \ \ \ \ x=3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 = (- \sqrt{\frac{1}{3}})^2}\\\\[/tex]
Therefore the final answer is "Option A".
Learn more:
brainly.com/question/15116572