Single polynomial, two remainder/factor conditions

\(f(x) = 6x^3 + px^2 + qx + 8\), where \(p\) and \(q\) are constants. Given that \(f(x)\) is exactly divisible by \((2x - 1)\), and also that when \(f(x)\) is divided by \((x - 1)\) the remainder is \(-7\),

1. find the value of \(p\) and the value of \(q\). [6]
2. Hence factorise \(f(x)\) completely. [3]

f(x) = ax³ + bx² - 7x + 14, where a and b are constants. Given that when f(x) is divided by (x - 1) the remainder is 9.

1. write down an equation connecting a and b. [2 marks] Given also that (x + 2) is a factor of f(x),
2. find the values of a and b. [4 marks]

\(f(x) = ax^3 + bx^2 - 7x + 14\), where \(a\) and \(b\) are constants. Given that when \(f(x)\) is divided by \((x - 1)\) the remainder is 9,

1. write down an equation connecting \(a\) and \(b\). [2]

Given also that \((x + 2)\) is a factor of \(f(x)\),

1. find the values of \(a\) and \(b\). [4]

The cubic polynomial \(f(x)\) is given by $$f(x) = x^3 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 1)\) is a factor of \(f(x)\) and that the remainder when \(f(x)\) is divided by \((x - 3)\) is 16.

1. Find the values of \(a\) and \(b\). [5]
2. Hence verify that \(f(2) = 0\), and factorise \(f(x)\) completely. [3]

\(f(x) = x^4 + bx + c\) \((x-2)\) is a factor of \(f(x)\). \(f(-3) = 35\).

1. Find \(b\) and \(c\). [4]
2. Hence express \(f(x)\) as the product of linear and cubic factors. [3]