Question

Determine whether the rational root theorem provides a complete list of all roots for the following polynomial functions.

Answer 1

Answer: 1) D.Yes.

2) A. No, this polynomial has complex roots.

3) C. No, this polynomial has irrational and complex roots.

Step-by-step explanation:

f(x)=$4x^2-25$ and to find the roots put f(x)=0

$\Rightarrow4x^2-25=0\\\Rightarrow4x^2=25\Rightarrow\ x^2=\frac{25}{4}\\\Rightarrow\ x=\frac{5}{2},\frac{-5}{2}$

where 2 is the factor of 4 and 5 is the factor of 25 .

∴ D.Yes, the rational root theorem provides a complete list of all roots for the f(x).

Now g(x)=$4x^2+25$ and to find the roots put g(x)=0

$\Rightarrow4x^2+25=0\\\Rightarrow4x^2=-25\Rightarrow\ x^2=\frac{-25}{4}\\\Rightarrow\ x=\frac{5i}{2},\frac{-5i}{2}$...> which are complex.

∴A. No, this polynomial has complex roots

h(x)=$3x^2+25$ and to find the roots put h(x)=0

$\Rightarrow3x^2+25=0\\\Rightarrow3x^2=-25\Rightarrow\ x^2=\frac{-25}{3}\\\Rightarrow\ x=\frac{5i}{\sqrt{3} },\frac{-5i}{\sqrt{3} }$...> which are irrational and complex .

∴C. No, this polynomial has irrational and complex roots

Answer 2

1)
f(x) = 4x^2 − 25 
f(x) = 4x^2 − 25 =0 implies  4x²=25, and x²=25/4, so the possible roots are 
x=sqrt(25/4= 5/2   or x= -sqrt(25/4)= -5/2
so the answer is D.Yes.

2)
g(x) = 4x^2 + 25=0 implies  4x²= -25 and since  i²= -1 (complex number)
we can write 4x²= -25= 25i² so the possible root is x= sqrt(25i²/4)=5i/2 or x= -sqrt(25i²/4)=-5i/2
the answer is A. No, this polynomial has complex roots
3)
h(x) = 3x^2 − 25=0 implies  3x²=25, x²=25/3 so the possible roots are x= -sqrt(25/3)= -5/sqrt3 or x=sqrt(25/3)=5/sqrt3
the answer is B. No, this polynomial has irrational roots