Algebra 2 help with polynomial equations and roots?

Question

A fifth degree polynomial equation with rational coefficients has the roots 2,-7i and -7+i.find the remaining roots
poss ans: a. 7i, -7+i B. -2, 7i, -7-i C. 7i, -7-i D. -7i, 7-i

A polynomial equation with rational coefficients has roots ! +sq rt 3 and 3 - sq rt 2. Find two additional roots.
Poss ans: A. 1- sq rt 3, 3 +sq rt 2
B. -1- sq rt 3, 3+ sq rt 2
C. -1+sq rt 3, -3+sq rt 2
d. -1-sq rt 3, -3+sq rt 2

Answer 1

Given that root are 2, -7i and -7 + i.

Here two roots are imaginary and complex.

By the conjugate pairs theorem (complex conjugate theorem) is i = -i.

So, taking conjugate of each imaginary root.

Conjugate of -7i is 7(-i) = 7i

Conjugate of -7+ i is -7 + (-i) = -7 - i.

Therefore remaining roots much be 7i and -7 - i.

Option C is right choice.

 

A polynomial equation with rational coefficients has roots 1+√3 and 3 - √2.

By using quadratic formula [-b±√(b²-4ac)]/2a

So, 1 - √3 and 3 + √2.

Therefore two additional roots are 1 - √3 and 3 + √2.

Option A is right choice.