Answer: The factors of [tex]x^3+5x^2+2x-8[/tex] are (x+4)(x-1) and (x+2).
Step-by-step explanation:
Given polynomial f(x)=[tex]x^3+5x^2+2x-8[/tex]
Let's check all the factors by applying Factor Theorem .
The Factor Theorem states that a polynomial f(x) has a factor (x-t) if and only if f(t)=0.
1. Let g(x)=x+5
here t=-5
Now applying factor theorem
f(t)=f(-5)=[tex]-5^3+5(-5)^2+2(5)-8=2[/tex]≠0
⇒ g(x)=x+5 is not a factor of f(x).
2.Let g(x)=x-3
here t=3
Now applying factor theorem
f(t)=f(3)=[tex]3^3+5(3)^2+2(3)-8=70[/tex]≠0
⇒g(x)=x-3 is not a factor f(x).
3.Let g(x)=x+4
here t=-4
Now applying factor theorem
f(t)=f(-4)=[tex]-4^3+5(-4)^2+2(-4)-8=0[/tex]
⇒g(x)=x+4 is a factor of f(x).
4. Let g(x)=x-1
here t=1
Now applying factor theorem
f(t)=f(1)=[tex]1^3+5(1)^2+2(1)-8=0[/tex]
⇒g(x)=x-1 is a factor of f(x).
5.Let g(x)=x+3
here t=-3
Now applying factor theorem
f(t)=f(-3)=[tex]-3^3+5(-3)^2+2(-3)-8=4[/tex]≠0
⇒ g(x) is not a factor of f(x).
6.Let g(x)=x+2
here t=-2
Now applying factor theorem
f(t)=f(-2)=[tex]-2^3+5(-2)^2+2(-2)-8=0[/tex]
⇒ g(x)=x+2 is the factor of f(x).
