Polynomial with equal remainders

3 The polynomial \(x ^ { 3 } + 4 x ^ { 2 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) is equal to the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and find the quotient when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\).

1 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 5 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x - 4 )\) the remainder is equal to 3 times the remainder when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).

5. $$f ( x ) = 3 x ^ { 3 } + A x ^ { 2 } + B x - 10$$ where \(A\) and \(B\) are integers.
Given that

• when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is \(k\)
• when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is \(- 10 k\)
• \(k\) is a constant
  1. show that

$$11 A + 9 B = 83$$ Given also that \(( 3 x - 2 )\) is a factor of \(\mathrm { f } ( x )\),

find the value of \(A\) and the value of \(B\).

Hence find the quadratic expression \(\mathrm { g } ( x )\) such that $$f ( x ) = ( 3 x - 2 ) g ( x )$$

6. $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. The remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) is equal to the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).

1. Find the value of \(a\). Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(b\).

7 The remainder when \(x ^ { 3 } - 2 x + 4\) is divided by ( \(x - 2\) ) is twice the remainder when \(x ^ { 2 } + x + k\) is divided by ( \(x + 1\) ).
Find the value of \(k\).

9. The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where \(k\) is a constant.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) the remainder is \(r\).
When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) the remainder is \(3 r\).

1. Find the value of \(k\).
2. Find the value of \(r\).
3. Show that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\).
4. Show that there is only one real solution to the equation \(\mathrm { f } ( x ) = 0\). END

f(x) = px³ + 6x² + 12x + q. Given that the remainder when f(x) is divided by (x - 1) is equal to the remainder when f(x) is divided by (2x + 1),

1. find the value of p. [4]

Given also that q = 3, and p has the value found in part (a),

1. find the value of the remainder. [1]

The cubic polynomial \(x^3 + ax^2 + bx - 6\) is denoted by f\((x)\).

1. The remainder when f\((x)\) is divided by \((x - 2)\) is equal to the remainder when f\((x)\) is divided by \((x + 2)\). Show that \(b = -4\). [3]
2. Given also that \((x - 1)\) is a factor of f\((x)\), find the value of \(a\). [2]
3. With these values of \(a\) and \(b\), express f\((x)\) as a product of a linear factor and a quadratic factor. [3]
4. Hence determine the number of real roots of the equation f\((x) = 0\), explaining your reasoning. [3]

The polynomial f(x) is given by $$\text{f}(x) = x^3 + kx^2 - 7x - 15,$$ where \(k\) is a constant. When f(x) is divided by \((x + 1)\) the remainder is \(r\). When f(x) is divided by \((x - 3)\) the remainder is \(3r\).

1. Find the value of \(k\). [5]
2. Find the value of \(r\). [1]
3. Show that \((x - 5)\) is a factor of f(x). [2]
4. Show that there is only one real solution to the equation f(x) = 0. [4]