Polynomial Factoring
The function \[f(x) = x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24\] has only four distinct roots, each of which is real. Let the four roots be \(\alpha,\ \beta,\ \gamma,\) and \(\delta,\) in no particular order. Also let \(f'(x)\) denote the first derivative of \(f(x).\) Evaluate \[f'(\alpha)\times f'(\beta)\times f'(\gamma)\times f'(\delta).\]
by
Caleb Townsend
The function \(f(x)=x^4+ax^3+bx^2+cx+d\) has four positive roots. We are also given that \((b-a)(a+b)+(d-c)(c+d)=2(ac-bd).\)
What is the value of \((a+b+c+d)^2?\)
by
mathh mathh
If \(a, b, c \) are real numbers such that \[a^2 + b^2 +2c^2 - 4a+2c -2bc +5 = 0 ,\] then what is the value of \(a+b-c\)?
by
Alamuru Ganesh
Let \(f(x) = x^3 + 2x^2 + 3x + 2\) and \(g(x)\) be a polynomial with integer coefficients. When \(f(x)\) is divided by \(g(x)\), it leaves quotient \(q(x)\) and remainder \(r(x)\), both of which have integer coefficients.
If \(q(x) = r(x) \neq 1\) ,then find \(g(5)\).
by
Nihar Mahajan
\[\large x^6 + x^5 + 2x^4 + 2x^3 + 3x^2 + 3x + 3 \]
How many real roots does the polynomial above have?
by
Rohit Kumar