Polynomial Factoring

Polynomial Factoring: Level 3 Challenges


The function f(x)=x5+6x418x310x2+45x24f(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 α, β, γ,\alpha,\ \beta,\ \gamma, and δ,\delta, in no particular order. Also let f(x)f'(x) denote the first derivative of f(x).f(x). Evaluate f(α)×f(β)×f(γ)×f(δ).f'(\alpha)\times f'(\beta)\times f'(\gamma)\times f'(\delta).

The function f(x)=x4+ax3+bx2+cx+df(x)=x^4+ax^3+bx^2+cx+d has four positive roots. We are also given that (ba)(a+b)+(dc)(c+d)=2(acbd).(b-a)(a+b)+(d-c)(c+d)=2(ac-bd).

What is the value of (a+b+c+d)2?(a+b+c+d)^2?

If a,b,ca, b, c are real numbers such that a2+b2+2c24a+2c2bc+5=0,a^2 + b^2 +2c^2 - 4a+2c -2bc +5 = 0 , then what is the value of a+bca+b-c?

Let f(x)=x3+2x2+3x+2f(x) = x^3 + 2x^2 + 3x + 2 and g(x)g(x) be a polynomial with integer coefficients. When f(x)f(x) is divided by g(x)g(x), it leaves quotient q(x)q(x) and remainder r(x)r(x), both of which have integer coefficients.

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

x6+x5+2x4+2x3+3x2+3x+3\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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