Question

Let P(x) = x⁶-2x⁵-x⁴+x³+2x²+x-2 (a) Determine whether (x−2) is a factor of P(x). (b) Find all the possible rational zeros of P(x) by using the Rational Zeros Theorem. (c) Proceed with any additional questions or tasks related to the polynomial P(x).

Answer 1

To determine if (x - 2) is a factor of P(x), we evaluate P(2) and find that P(2) = 0, so (x - 2) is indeed a factor. The possible rational zeros of P(x) are ±2, and ±1, as determined by the Rational Zeros Theorem. Additional tasks would involve factoring P(x) and finding other roots, using algebraic techniques.

Step-by-step explanation:

To determine whether (x−2) is a factor of P(x), we can apply the Remainder Theorem which states that if a polynomial P(x) is divided by (x−a), the remainder is P(a). So, we'll evaluate P(2):

P(2) = 26 - 2⋅25 - 24 + 23 + 2⋅22 + 2 - 2

P(2) = 64 - 64 - 16 + 8 + 8 + 2 - 2

P(2) = 0

Since P(2) is 0, (x−2) is a factor of P(x).

For the Rational Zeros Theorem, the possible rational zeros are all the factors of the constant term divided by the factors of the leading coefficient. In P(x), the constant term is -2, and the leading coefficient is 1, so the possible rational zeros are ±2, and ±1. It's important to note that these are just possible zeros; they need to be tested to see if they're actual zeros of the polynomial.

Further Exploration of P(x)

If there are additional tasks or steps needed like factoring P(x), synthetic division, or finding other roots, we would proceed with the appropriate algebraic techniques to fully analyze the polynomial P(x).