Mathematics

Question

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

f(x) = 4x^2 − 25
A. No, this polynomial has complex roots
B. No, this polynomial has irrational roots
C. No, this polynomial has irrational and complex roots
D.Yes.

g(x) = 4x^2 + 25
A. No, this polynomial has complex roots
B. No, this polynomial has irrational roots
C. No, this polynomial has irrational and complex roots
D.Yes.

h(x) = 3x^2 − 25
A. No, this polynomial has complex roots
B. No, this polynomial has irrational roots
C. No, this polynomial has irrational and complex roots
D.Yes.

1 Answer

  • 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
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