Identities and inequalities: แบบฝึกหัด

Solve the inequality \(x^2 - 2x - 15 \le 0\).

The polynomial \(f(x) = x^3 + px^2 + qx + 6\) is divisible by \((x - 1)\) and leaves a remainder of \(12\) when divided by \((x - 2)\).
Find the value of \(p\) and the value of \(q\).

The polynomial \( f(x) = x^3 + ax^2 + bx + 6 \) has a factor \( (x - 1) \). When \( f(x) \) is divided by \( (x + 2) \), the remainder is \( -12 \). Find the values of \( a \) and \( b \).

Find the remainder when \(f(x) = 3x^3 - 2x^2 + 5x - 4\) is divided by \((x - 1)\).

Find the set of values of \( x \) for which \( 2x^2 - 5x - 3 > 0 \) and \( 4x + 1 > 2x + 7 \) are both satisfied.

The polynomial \(f(x) = 3x^3 - 2x^2 + kx - 8\) leaves a remainder of \(10\) when divided by \((x - 2)\).
Find the value of the constant \(k\).

Find the set of values of \( x \) for which \( 2x^2 - 5x - 3 > 0 \).

Solve the inequality \(x^2 - 2x - 15 > 0\).

The polynomial \(f(x) = x^3 - 4x^2 + kx + 6\) leaves a remainder of \(12\) when divided by \((x - 3)\).
(a) Find the value of the constant \(k\).
(b) Using your value of \(k\), show that \((x - 1)\) is a factor of \(f(x)\).

The function \( f(x) = 2x^3 + ax^2 + bx - 6 \) has a factor \( (x - 2) \). When \( f(x) \) is divided by \( (x + 1) \), the remainder is \( -12 \).
(a) Find the value of \( a \) and the value of \( b \).
(b) Factorise \( f(x) \) completely.
(c) Solve the inequality \( f(x) > 0 \).
(d) On a set of axes, sketch the region representing the solution to the simultaneous inequalities \( y < f(x) \) and \( y > 0 \).