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–√)
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))
To determine the remaining two roots of the polynomial P(x), we need to use the fact that the sum of the roots of a polynomial is equal to the negative coefficient of the second-to-last term (when the polynomial is written in descending order).
In this case, the coefficient of the second-to-last term is 226, so the sum of the roots is -226. We already know three of the roots are -1, 2-√3, and 1+√2. Let's call the remaining two roots A and B.
We can set up an equation to find the sum of the remaining roots:
-1 + (2-√3) + (1+√2) + A + B = -226.
Simplifying, we get:
2 - √3 + 1 + √2 + A + B = -226.
Combining like terms, we have:
A + B - √3 + √2 = -229.
Since A and B have to be real numbers, the terms involving square roots must cancel each other out. This means √3 must be equal to -√2. Therefore:
A + B - √3 + √2 = -229
A + B - √2 - √2 = -229
A + B - 2√2 = -229.
Now, we have two equations to find the values of A and B:
A + B = -229 (from the first equation)
A + B - 2√2 = -229 (from the second equation)
Solving this system of equations, we find that A = 1 - 2√3 and B = 3√2 - 1.
Therefore, the remaining two roots of P(x) are 1 - 2√3 and 3√2 - 1.
Now, let's move on to Part B to determine the correct fully factored form of P(x).
Looking at the given options, we can see that option D, p(x) = (x+1)(x-1-√23)(x-2+3√2), is the only one that matches the roots we found.
Therefore, the correct fully factored form of P(x) is p(x) = (x+1)(x-1-√23)(x-2+3√2).
So, your answers are correct:
Part A: The remaining two roots of P(x) are 1 - 2√3 and 3√2 - 1.
Part B: The fully factored form of P(x) is p(x) = (x+1)(x-1-√23)(x-2+3√2).