代數與函數：练习题

Simplify the rational expression \(\frac{2x^2 + 5x - 3}{x^2 + 2x - 3}\) for all values where the denominator is non-zero.

The rational function \(f(x) = \frac{3x+5}{(x-1)^2(x+2)}\) can be decomposed into partial fractions as:
\(\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+2}\).
Find the value of the constant \(A\).

The quadratic equation \(x^2 + (k+2)x + (2k+1) = 0\) has no real roots. Find the range of possible values for the constant \(k\).

Given that \(f(x) = x^3 + ax^2 + bx - 6\), \((x-1)\) is a factor of \(f(x)\). When \(f(x)\) is divided by \((x+1)\), the remainder is \(-10\). Calculate the value of \(a^2 + b^2\).

The polynomial \(f(x)\) is defined by \(f(x) = 2x^3 + ax^2 - 5x + 6\), where \(a\) is a constant. Given that \((x - 2)\) is a factor of \(f(x)\), find the value of \(a\).

The polynomial \(f(x) = 2x^3 - 5x^2 + ax + b\) is divisible by \((x + 1)\). When \(f(x)\) is divided by \((x - 2)\), the remainder is 7. Find the values of the constants \(a\) and \(b\).

Determine the coordinates of the points of intersection between the graph of the modulus function \(y = |3x - 6|\) and the straight line with equation \(y = x + 2\).

Find the set of values of \(k\) for which the quadratic equation \(kx^2 + 4x + (k-3) = 0\) has real roots.

The polynomial \(f(x)\) is defined by \(f(x) = x^3 + px^2 - 11x + q\), where \(p\) and \(q\) are constants.
(a) Given that \((x - 2)\) is a factor of \(f(x)\) and that the remainder when \(f(x)\) is divided by \((x + 1)\) is 6, find the values of \(p\) and \(q\).
(b) Factorize \(f(x)\) completely into linear factors.