Algebra：练习题

Given that \((x - 1)\) is a factor of the polynomial \(P(x) = x^3 - 3x^2 + kx - 2\), find the value of the constant \(k\).

Solve the inequality \(x^2 - 5x + 6 < 0\).

Find the first three terms in the expansion of \((1 - 2x)^{-2}\) in ascending powers of \(x\), where \(|2x| < 1\).

A polynomial is given by \(P(x) = 2x^3 - 3x^2 + ax + b\). Given that \((x - 2)\) is a factor of \(P(x)\) and the remainder when \(P(x)\) is divided by \((x + 1)\) is \(-12\), find the values of the constants \(a\) and \(b\).

The polynomial \(P(x) = ax^3 + 7x^2 - 11x + b\) has a factor \((x - 2)\). When \(P(x)\) is divided by \((x + 1)\), the remainder is 12. Find the values of the constants \(a\) and \(b\).

Given that \((x + 3)\) is a factor of the polynomial \(P(x) = x^3 + kx^2 - 7x + 6\), find the value of the constant \(k\).

Find the quotient and the remainder when the polynomial \(2x^3 - 3x^2 + 4x + 1\) is divided by \(x^2 - 1\).

Find the first three terms, in ascending powers of \(x\), of the expansion of \((1 - 2x)^{-2}\), stating the range of values of \(x\) for which the expansion is valid.

A polynomial is defined by \(P(x) = ax^3 + 4x^2 + bx - 1\).

(a) When \(P(x)\) is divided by \((x-1)\), the remainder is 4.

(b) When \(P(x)\) is divided by \((x+2)\), the remainder is -11.

Find the values of the constants \(a\) and \(b\).

(a) Solve the equation \(|2x - 3| = |x + 4|\).
(b) Hence, using your answers from part (a), solve the inequality \(|2x - 3| < |x + 4|\).