SIMAK UI · Basic Mathematics (Matematika Dasar)

Algebra, Polynomials and Rational Inequalities

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

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If the polynomial $P (x) = x^{3} - a x^{2} + b x - 6$ is divisible by $(x - 1)$ and leaves a remainder of 8 when divided by $(x - 3)$ , find the value of $2 a + b$ .
Find all real solutions to the rational inequality $\frac{x ^{2} - 4 x + 3}{x - 2} \leq 0$ .
If $lo g_{2} (x - 1) + lo g_{2} (x + 2) = 2$ , find the real value of $x$ .
If the roots of the quadratic equation $2 x^{2} - 6 x + k = 0$ satisfy $α^{2} + β^{2} = 5$ , find the value of the parameter $k$ .
Solve the radical equation $2 x + 7 - x = 2$ for all real roots $x$ .
If $x + \frac{1}{x} = 4$ , determine the numerical value of $x^{3} + \frac{1}{x ^{3}}$ .
Determine the set of all real values of $x$ satisfying the absolute value inequality $∣3 x - 5∣ \geq 2 x + 1$ .
If the system of linear equations:
$a x + 2 y = 4$
$3 x + (a - 1) y = 6$
has infinitely many solutions, what is the value of the constant $a$ ?
Find the sum of all real roots of the biquadratic equation $x^{4} - 13 x^{2} + 36 = 0$ .
If $2^{2 x + 1} - 9 (2^{x}) + 4 = 0$ , find the complete set of solutions for $x$ .
The polynomial $f (x) = 2 x^{3} + p x^{2} + q x - 2$ has $(x - 2)$ and $(x + 1)$ as factors. Find the product $p \times q$ .
Determine the solution set of the system:
$\frac{1}{x} + \frac{1}{y} = \frac{5}{6}$
$\frac{1}{x} - \frac{1}{y} = \frac{1}{6}$
For what values of $m$ does the quadratic equation $x^{2} - (m + 2) x + (2 m + 1) = 0$ have real and equal roots?
If $lo g_{3} 2 = a$ and $lo g_{3} 5 = b$ , express $lo g_{3} 50$ in terms of $a$ and $b$ .
Solve the inequality $\frac{2 x - 1}{x + 3} \geq 1$ .
If $(x + y) = 5$ and $x^{2} + y^{2} = 13$ , find the value of $x^{4} + y^{4}$ .
Find all real solutions $x$ satisfying $3^{2 x} - 10 (3^{x}) + 9 = 0$ .
What is the domain of the real-valued function $f (x) = \frac{x - 3}{2 - x}$ ?
If the polynomial $x^{3} - k x^{2} + 11 x - 6 = 0$ has three roots in arithmetic progression, find the value of $k$ .
Determine the remainder when $x^{2024} - 2 x + 1$ is divided by $(x - 1)$ .
Solve the logarithmic inequality $lo g_{1/2} (2 x - 3) > - 2$ .
If $\frac{x}{y} + \frac{y}{x} = 3$ , what is the value of $\frac{x ^{2}}{y ^{2}} + \frac{y ^{2}}{x ^{2}}$ ?
Find all real values of $x$ such that $∣ x - 2∣ + ∣ x - 5∣ = 7$ .
If $α$ and $β$ are roots of $x^{2} - 4 x + 2 = 0$ , construct the quadratic equation whose roots are $α + 1$ and $β + 1$ .
Determine the number of distinct real solutions to the equation $x^{2} - 5∣ x ∣ + 6 = 0$ .

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Algebra, Polynomials and Rational Inequalities

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