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.

Question

Mathematics

Mikaela Harris

24 October, 22:41

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.

+2

Answers (1)

Know the Answer?

Imani Butler · Answer 1

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