The polynomial \(2x^3 + ax^2 + bx + 6\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). When \(p(x)\) is divided by \((x + 2)\) the remainder is \(-38\) and when \(p(x)\) is divided by \((2x - 1)\) the remainder is \(\frac{19}{2}\).

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

The polynomial \(ax^3 + 5x^2 - 4x + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x + 2)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x + 1)\) the remainder is 2.

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

Find the quotient and remainder when \(6x^4 + x^3 - x^2 + 5x - 6\) is divided by \(2x^2 - x + 1\).

The polynomial \(6x^3 + ax^2 + bx - 2\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x + 2)\) the remainder is \(-24\). Find the values of \(a\) and \(b\).

The polynomial \(x^4 + 3x^3 + ax + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). When \(p(x)\) is divided by \(x^2 + x - 1\) the remainder is \(2x + 3\). Find the values of \(a\) and \(b\).

The polynomial \(x^4 + 2x^3 + ax + b\), where \(a\) and \(b\) are constants, is divisible by \(x^2 - x + 1\). Find the values of \(a\) and \(b\).

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

The polynomial \(4x^4 + ax^2 + 11x + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \(x^2 - x + 2\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the real roots of the equation \(p(x) = 0\).

The polynomial \(4x^3 + ax + 2\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\).

1. Find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(p(x)\),
   2. solve the inequality \(p(x) > 0\), justifying your answer.

Show that \((x + 1)\) is a factor of \(4x^3 - x^2 - 11x - 6\).

The polynomial \(8x^3 + ax^2 + bx - 1\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((2x + 1)\) the remainder is 1.

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

(ii) When \(a\) and \(b\) have these values, factorise \(p(x)\) completely.

The polynomial \(2x^3 + ax^2 - 11x + b\) is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \((2x - 1)\) and that when \(p(x)\) is divided by \((x + 1)\) the remainder is 12.

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

The polynomial \(4x^3 + ax^2 + bx - 2\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x+1)\) and \((x+2)\) are factors of \(p(x)\).

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

(ii) When \(a\) and \(b\) have these values, find the remainder when \(p(x)\) is divided by \((x^2 + 1)\).

The polynomial \(ax^3 + bx^2 + x + 3\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((3x + 1)\) is a factor of \(p(x)\), and that when \(p(x)\) is divided by \((x - 2)\) the remainder is 21. Find the values of \(a\) and \(b\).

The polynomial \(f(x)\) is defined by

\(f(x) = x^3 + ax^2 - ax + 14\),

where \(a\) is a constant. It is given that \((x + 2)\) is a factor of \(f(x)\).

(i) Find the value of \(a\).

(ii) Show that, when \(a\) has this value, the equation \(f(x) = 0\) has only one real root.

The polynomial \(8x^3 + ax^2 + bx + 3\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((2x - 1)\) the remainder is 1.

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(p(x)\) is divided by \(2x^2 - 1\).

The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((3x + 1)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, factorise \(p(x)\) completely.

Find the quotient and remainder when \(2x^2\) is divided by \(x + 2\).

The polynomial \(p(x)\) is defined by

\(p(x) = x^3 - 3ax + 4a\),

where \(a\) is a constant.

(i) Given that \((x - 2)\) is a factor of \(p(x)\), find the value of \(a\).

(ii) When \(a\) has this value,

(a) factorise \(p(x)\) completely,

(b) find all the roots of the equation \(p(x^2) = 0\).

The polynomial \(p(x)\) is defined by

\(p(x) = ax^3 - x^2 + 4x - a\),

where \(a\) is a constant. It is given that \((2x - 1)\) is a factor of \(p(x)\).

Find the value of \(a\) and hence factorise \(p(x)\).

The polynomial \(x^4 + 3x^3 + ax + 3\) is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \(x^2 - x + 1\).

1. Find the value of \(a\).
2. When \(a\) has this value, find the real roots of the equation \(p(x) = 0\).

The polynomial \(ax^3 + bx^2 + 5x - 2\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x - 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x - 2)\) the remainder is 12.

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

(ii) When \(a\) and \(b\) have these values, find the quadratic factor of \(p(x)\).

Find the quotient and remainder when \(2x^4 - 27\) is divided by \(x^2 + x + 3\).

The polynomial \(f(x)\) is defined by

\(f(x) = 12x^3 + 25x^2 - 4x - 12\).

(i) Show that \(f(-2) = 0\) and factorise \(f(x)\) completely.

(ii) Given that

\(12 \times 27^y + 25 \times 9^y - 4 \times 3^y - 12 = 0\),

state the value of \(3^y\) and hence find \(y\) correct to 3 significant figures.

The polynomial \(p(z)\) is defined by

\(p(z) = z^3 + mz^2 + 24z + 32\),

where \(m\) is a constant. It is given that \((z + 2)\) is a factor of \(p(z)\).

1. Find the value of \(m\).
2. Hence, showing all your working, find
   1. the three roots of the equation \(p(z) = 0\),
   2. the six roots of the equation \(p(z^2) = 0\).

The polynomial \(2x^3 + 5x^2 + ax + b\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((2x + 1)\) is a factor of \(p(x)\) and that when \(p(x)\) is divided by \((x + 2)\) the remainder is 9.

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

(ii) When \(a\) and \(b\) have these values, factorise \(p(x)\) completely.

The polynomial \(4x^3 - 4x^2 + 3x + a\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \(2x^2 - 3x + 3\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, solve the inequality \(p(x) < 0\), justifying your answer.

The polynomial \(x^4 + 3x^2 + a\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \(x^2 + x + 2\) is a factor of \(p(x)\). Find the value of \(a\) and the other quadratic factor of \(p(x)\).

The polynomial \(x^3 - 2x + a\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((x + 2)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, find the quadratic factor of \(p(x)\).

The polynomial \(x^4 + 5x + a\) is denoted by \(p(x)\). It is given that \(x^2 - x + 3\) is a factor of \(p(x)\).

(i) Find the value of \(a\) and factorise \(p(x)\) completely.

(ii) Hence state the number of real roots of the equation \(p(x) = 0\), justifying your answer.

The polynomial \(2x^3 + ax^2 - 4\) is denoted by \(p(x)\). It is given that \((x - 2)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

When \(a\) has this value,

(ii) factorise \(p(x)\),

(iii) solve the inequality \(p(x) > 0\), justifying your answer.

The polynomial \(x^4 - 2x^3 - 2x^2 + a\) is denoted by \(f(x)\). It is given that \(f(x)\) is divisible by \(x^2 - 4x + 4\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, show that \(f(x)\) is never negative.

The polynomial \(x^4 + 4x^2 + x + a\) is denoted by \(p(x)\). It is given that \((x^2 + x + 2)\) is a factor of \(p(x)\).

Find the value of \(a\) and the other quadratic factor of \(p(x)\).

The polynomial \(2x^4 + ax^3 + bx - 1\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). When \(p(x)\) is divided by \(x^2 - x + 1\) the remainder is \(3x + 2\).

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

The polynomial \(ax^3 + x^2 + bx + 3\) is denoted by \(p(x)\). It is given that \(p(x)\) is divisible by \((2x - 1)\) and that when \(p(x)\) is divided by \((x + 2)\) the remainder is 5.

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

The polynomial \(ax^3 - 10x^2 + bx + 8\), where \(a\) and \(b\) are constants, is denoted by \(p(x)\). It is given that \((x-2)\) is a factor of both \(p(x)\) and \(p'(x)\).

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

(b) When \(a\) and \(b\) have these values, factorise \(p(x)\) completely.

Find the quotient and remainder when \(8x^3 + 4x^2 + 2x + 7\) is divided by \(4x^2 + 1\).

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