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Polynomial Factoring

A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

Level 3

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 \[f'(\alpha)\times f'(\beta)\times f'(\gamma)\times f'(\delta).\]

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 \((a+b+c+d)^2?\)

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 \(a+b-c\)?

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 \(r(x)\).

If \(q(x) = r(x) \neq 1\) ,then find \(g(5)\).

\[\large x^6 + x^5 + 2x^4 + 2x^3 + 3x^2 + 3x + 3 \]

How many real roots does the polynomial above have?

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