Respuesta :
Answer:
All potential roots are 3,3 and [tex]-\frac{5}{3}[/tex].
Step-by-step explanation:
Potential roots of the polynomial is all possible roots of f(x).
[tex]f(x)=3x^3-13x^2-3x+45[/tex]
Using rational root theorem test. We will find all the possible or potential roots of the polynomial.
p=All the positive/negative factors of 45
q=All the positive/negative factors of 3
[tex]p=\pm 1,\pm 3,\pm 5\pm \pm 9,\pm 15\pm 45[/tex]
[tex]q=\pm 1,\pm 3[/tex]
All possible roots
[tex]\frac{p}{q}=\pm 1,\pm 3,\pm 5\pm \pm 9,\pm 15\pm 45,\pm \frac{1}{3},\pm \frac{5}{3}[/tex]
Now we check each rational root and see which are possible roots for given function.
[tex]f(1)= 3\times 1^3-13\times 1^2-3\times 1+45\Rightarrow 32\neq 0[/tex]
[tex]f(-1)= 3\times (-1)^3-13\times (-1)^2-3\times (-1)+45\Rightarrow \neq 32[/tex]
[tex]f(-3)= 3\times (-3)^3-13\times (-3)^2-3\times (-3)+45\Rightarrow \neq -144[/tex]
[tex]f(3)= 3\times (3)^3-13\times (3)^2-3\times (3)+45\Rightarrow =0\\\\ \therefore x=3\text{ Potential roots of function}[/tex]
Similarly, we will check for all value of p/q and we get
[tex]f(-5/3)=0[/tex]
Thus, All potential roots are 3,3 and [tex]-\frac{5}{3}[/tex].
The potential roots of the expression [tex]\rm f(x) = 3x^3 -13x^2 -3x + 45[/tex] are 3,3, and -5/3.
Given
The given expression is;
[tex]\rm f(x) = 3x^3 -13x^2 -3x + 45[/tex]
What are potential roots?
The Potential roots are defined as the factors of the constant term by the factors of the leading coefficient.
The following formula is used to calculate potential roots;
[tex]\rm \text{Potential roots} = \dfrac{Factors\ of \ the \ constant \ term }{ Factors \ of \ leading \ coefficient}[/tex]
The potential roots of expression are;
[tex]\rm p=\pm1, \pm 3,\pm 5, \pm9, \pm 15, \pm45\\\\q=\pm1, \pm3[/tex]
The common possible roots are [tex]\rm \pm1, \pm 3[/tex].
Therefore,
Substitute all common possible roots in the equation
[tex]\rm f(x) = 3x^3 -13x^2 -3x + 45\\\\f(1)=3(1)^3-13(1)^2-3(1)+45= 3-13-3+45=32\neq 0\\\\f(-1)=3(-1)^3-13(-1)^2-3(-1)+45= -3-13+3+45=32\neq 0\\\\f(3)=3(3)^3-13(3)^2-3(3)+45= 81-117+9+45=0\\\\f(-3)=3(-3)^3-13(-3)^2-3(-3)+45=- 81-117+9+45=-144\neq0[/tex]
Here, x= 3 is the potential root of the given expression.
Hence, the potential roots of the expression [tex]\rm f(x) = 3x^3 -13x^2 -3x + 45[/tex] are 3,3, and -5/3.
To know more about Potential roots click the link given below.
https://brainly.com/question/11015301
