Riigieksamid (Estonian State Exams) · Mathematics

Advanced Algebra and Polynomials

11 practice questions, each with the answer and an explanation.

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What are the real roots of the polynomial equation $x^{3} - 4 x^{2} - x + 4 = 0$ ?
Solve the rational inequality $\frac{x + 2}{x - 5} \geq 0$ for real numbers $x$ .
What is the remainder when the polynomial $P (x) = 2 x^{3} - 5 x^{2} + 3 x - 7$ is divided by $x - 2$ ?
Find all real solutions to the logarithmic equation $lo g_{2} (x + 3) + lo g_{2} (x - 3) = 4$ .
Solve the exponential equation $5^{2 x} - 26 \cdot 5^{x} + 25 = 0$ .
For which values of $m$ does the quadratic equation $x^{2} + 2 m x + (m + 6) = 0$ have two distinct real roots?
What is the sum of the first 20 terms of an arithmetic progression whose first term is 5 and common difference is 4?
Find the sum to infinity of the geometric series $18 + 6 + 2 + \frac{2}{3} + \dots$ .
Simplify the algebraic fraction $\frac{x ^{3} + 8}{x ^{2} - 2 x + 4}$ for all real $x$ .
What is the coefficient of $x^{4}$ in the binomial expansion of $(2 x - 1)^{6}$ ?
Find the solution set of $∣2 x - 5∣ < 7$ in $R$ .

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