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Algebra

# Polynomial Factoring: Level 3 Challenges

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 $$r(x)$$, both of which have integer coefficients.

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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