Question

If ax^3 – bx^2 + c with a ≠ 0 and c ≠ 0 has a factor of the form x^2 – nx + 1, determine c^2 - a^2.

A) c^2 - a^2 = b^2 - 1
B) c^2 - a^2 = b^2
C) c^2 - a^2 = 1
D) c^2 - a^2 = b^2 + 1

Answer 1

For the given polynomial with the factor x^2 − nx + 1, the correct answer is A) c^2 − a^2 = b^2 − 1, determined by comparing coefficients after expansion of the factorized form. The correct answer from the given options is: A) c2 − a2 = b2 − 1

Step-by-step explanation:

If ax3 − bx2 + c with a ≠ 0 and c ≠ 0 has a factor of the form x2 − nx + 1, we can use polynomial division or factoring to determine a relationship between a, b, and c. Without loss of generality, let us factor the cubic polynomial assuming x2 − nx + 1 is a factor. The cubic polynomial can be express as the product of the quadratic factor and a linear factor ax: (x2 − nx + 1)(ax). Expanding this product gives us: ax3 − anx2 + ax. In order for this to match the original polynomial ax3 − bx2 + c, the coefficients must be equal, so a must equal 1, −an must equal −b, and a must equal c.

Comparing the coefficients gives us the following relationships:

• a = 1
• n = b/a
• c = a

Substituting these into c2 − a2 we get:

(a2) − (12) = b2 − 1

Therefore, c2 − a2 = b2 − 1

The correct answer from the given options is:

A) c2 − a2 = b2 − 1