Single polynomial, two remainder/factor conditions

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

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 5 x ^ { 3 } + a x ^ { 2 } + b x - 16$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 27 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer. \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
5. Find the exact area of the shaded region.

2 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is - 5 . Find the values of \(a\) and \(b\).

4 The cubic polynomial \(2 x ^ { 3 } - 5 x ^ { 2 } + a x + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 6 . Find the values of \(a\) and \(b\).

7 The polynomial \(3 x ^ { 3 } + 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 10 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, solve the equation \(\mathrm { p } ( x ) = 0\).

7 The polynomial \(a x ^ { 3 } - 3 x ^ { 2 } - 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 37 x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\). It is also given that the remainder is 40 when \(\mathrm { p } ( x )\) is divided by ( \(2 x - 1\) ).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

3 The polynomial \(a x ^ { 3 } + x ^ { 2 } + b x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by ( \(2 x - 1\) ) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 11 x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(( 2 x - 1 )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 12 . Find the values of \(a\) and \(b\).

10. $$f ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 5$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) there is no remainder.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is - 15

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

6. $$f ( x ) = a x ^ { 3 } - 8 x ^ { 2 } + b x + 6$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) there is no remainder. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 12

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

1. \(\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

When \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 1 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 28 .

1. Find the value of \(a\) and the value of \(b\).
2. Show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).

4. \(\mathrm { f } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } + b x + 4\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ) the remainder is 55
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 9

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

8 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 10\) is denoted by \(\mathrm { f } ( x )\). It is given that, when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 12 . It is also given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).

1. Find the values of \(a\) and \(b\).
2. Divide \(\mathrm { f } ( x )\) by ( \(x + 2\) ) to find the quotient and the remainder.

8 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + a x + c\) and that \(\mathrm { f } ( 2 ) = 11\). The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 8 . Find the values of \(a\) and \(c\).

6 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + 15$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 3\) ) is a factor of \(\mathrm { f } ( x )\) and that, when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 35 .

1. Find the values of \(a\) and \(b\).
2. Using these values of \(a\) and \(b\), divide \(\mathrm { f } ( x )\) by ( \(x + 3\) ).

2. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), the remainder is 14 . When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .

1. Find the value of \(a\) and the value of \(b\).
2. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
   [0pt] [P3 June 2002 Question 1]

6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,

1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
2. find the values of \(a\) and \(b\),
3. fully factorise \(\mathrm { p } ( x )\).

1

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2\).
   1. Find f(1).
   2. Show that \(\mathrm { f } ( - 2 ) = 0\).
   3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where \(a\) and \(b\) are integers.
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(( 3 x - 1 )\), the remainder is 2 . Find the value of \(d\).

The polynomial \(p(x)\) is defined by $$p(x) = ax^3 - ax^2 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 2)\) is a factor of \(p(x)\), and that the remainder is 35 when \(p(x)\) is divided by \((x - 3)\).

1. Find the values of \(a\) and \(b\). [5]
2. Hence factorise \(p(x)\) and show that the equation \(p(x) = 0\) has exactly one real root. [3]

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 \(b\). [5]

$$f(x) = Ax^3 + 6x^2 - 4x + B$$ where \(A\) and \(B\) are constants. Given that

• \((x + 2)\) is a factor of \(f(x)\)
• \(\int_{-3}^{5} f(x)dx = 176\)

Find the value of \(A\) and the value of \(B\). [7]

$$f(x) = ax^3 + bx^2 - 7x + 14, \text{ where } a \text{ and } b \text{ are constants.}$$ 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 function \(f(x) = x^4 + bx + c\) is such that \(f(2) = 0\). Also, when \(f(x)\) is divided by \(x + 3\), the remainder is \(85\). Find the values of \(b\) and \(c\). [5]