Question

If \( x^2 - 5x + 6 \) is a factor of \( f(x) = x^4 - 17x^3 + kx^2 - 247x + 210 \), find the other quadratic factor of \( f(x) \).

• A. \( x^2 + 12x + 35 \)
• B. \( x^2 - 12x + 35 \)
• C. \( x^2 - 6x + 35 \)
• D. \( x^2 + 6x + 35 \)

Hint:

Use polynomial factorization and coefficient comparison to find unknown factors.

Answer 1

The Correct Option is B

Assume \[ f(x) = (x^2 - 5x + 6)(x^2 + px + q). \]
Expand: \[ x^4 + p x^3 + q x^2 - 5x^3 - 5p x^2 - 5q x + 6 x^2 + 6 p x + 6 q. \]
Collect like terms: \[ x^4 + (p - 5) x^3 + (q - 5p + 6) x^2 + (-5q + 6p) x + 6 q. \]
Compare with \[ x^4 - 17x^3 + k x^2 - 247 x + 210, \]
which gives: \[ p - 5 = -17 \implies p = -12, \] \[ 6 q = 210 \implies q = 35. \]
Thus, the other quadratic factor is \( x^2 - 12x + 35 \).