Question

If q1(x)=x²+(k-29)x-k and q2(x)=2x²+(3k-43)x+k are both factors of a cubic polynomial then find the largest value of k

Answer 1

It is given that Q₁(x) = x² + (k - 29)x - k and Q₂(x) = 2x² + (2k - 43)x + k are both factors of a cubic polynomial.

To find : the largest value of k.

solution : here Q₁(x) and Q₂(x) must have a root in common for them to both be factors of the same cubic polynomial.

let's find the common root a.

Q₂(x) - 2Q₁(x) = 2x² + (2k - 43)x + k - 2{x² + (k - 29)x - k} = 0

⇒2x² + (2k - 43)x + k - 2x² - 2(k - 29)x + 2k = 0

⇒(2k - 43 - 2k + 58)x + 3k = 0

⇒(15)x + 3k = 0

⇒x = -k/5

hence x = -k/5 is a common root of both quadratic polynomials.

let's put x = -k/5 in Q₁(x),

Q₁(-k/5) = (-k/5)² + (k - 29)(-k/5) - k = 0

⇒k²/25 - k²/5 + 29k/5 - k = 0

⇒-4k²/25 + 24k/5 = 0

⇒-k²/25 + 6k/5 = 0

⇒-k² + 30k = 0

⇒k = 30, 0

Therefore the largest value of k = 30