Question

Hoa wants to factor the polynomial P(x)=x5−5x4−23x3+55x2+226x+154. She finds three roots of P(x), −1, 2−32–√, and 1+23–√.Complete Hoa’s work:

Part A: What are the remaining two roots of P(x)?
1
2+3√2
3√2-1
1-2√3

Part B: Which equation correctly represents the fully factored form of P(x)?
p(x)=(x+1)(x−1)(x−1−23–√)(x−2+32–√)
p(x)=(x−1)(x−1−23–√)(x−1+23–√)(x−2−32–√)(x−2+32–√)
p(x)=(x+1)(x−1−23–√)(x−1+23–√)(x−2−32–√)(x−2+32–√)
p(x)=(x+1)(x−1−23–√)(x−32–√+1)(x−2+32–√)

I think that A is 1-2√3
I think that B is p(x)=(x+1)(x−1)(x−1−23–√)(x−2+32–√)

Answer 1

You really should proofread your post.
since roots come in quadratic conjugate pairs,
If 2-3√2 is a root, so is 2+3√2
If 1+2√3 is a root, so is 1-2√3
p(x) is a 5th degree polynomial. Rational roots need not come i pairs, so you do not need 1.
So, p(x) = (x+1)(x-(2-3√2))(x-(2+3√2))(x-(1+2√3))(x-(1-2√3))