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The polynomial n(x) = 3x4 - 9x3 + x2 – 3x has a factor of (x – 3).
Complete each of the 2 activities for this Task.
Activity 1 of 2
Part A: State the number of roots of the polynomial n(x).
A. 2 real roots; 2 complex roots
B. 4 real roots
O C. 3 real roots; 1 complex root
0
D. 1 real root; 3 complex roots

The Polynomial Nx 3x4 9x3 X2 3x Has A Factor Of X 3 Complete Each Of The 2 Activities For This Task Activity 1 Of 2 Part A State The Number Of Roots Of The Poly class=

Sagot :

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

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