Factor & Remainder Theorem

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]

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.

$$f(x) = Ax^3 + 6x^2 - 4x + B$$ 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]

2. $$f ( x ) = ( 2 x - 3 ) ( x - k ) - 12$$ where \(k\) is a constant.
a.Write down the value of \(\mathrm { f } ( k )\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) the remainder is - 5
b. find the value of \(k\).
c. Factorise \(\mathrm { f } ( x )\) completely.

2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 72 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .

1. Find the values of \(a\) and \(b\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
   The curves meet at the point \(( 2,3 )\).
   Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
   Region \(B\) is bounded by the two curves and the straight line \(x = 0\).

1. Show that \((x + 2)\) is a factor of \(f(x) = x^3 - 19x - 30\). [2]
2. Factorise \(f(x)\) completely. [4]

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 3 x\).

1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
3. 1. Use the Remainder Theorem to find the remainder, \(r\), when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
   2. Using algebraic division, or otherwise, express \(\mathrm { p } ( x )\) in the form $$( x - 2 ) \left( x ^ { 2 } + a x + b \right) + r$$ where \(a , b\) and \(r\) are constants.

1. It is given that $$P(x) = 4x^3 + 8x^2 + 11x + 4$$ Use the factor theorem to show that \((2x + 1)\) is a factor of \(P(x)\) [2 marks]
2. Express \(P(x)\) in the form $$P(x) = (2x + 1)(ax^2 + bx + c)$$ where \(a\), \(b\) and \(c\) are constants to be found. [2 marks]
3. Given that \(n\) is a positive integer, use your answer to part (b) to explain why \(4n^3 + 8n^2 + 11n + 4\) is never prime. [2 marks]

$$\text{f}(x) = 2x^3 - 8x^2 + 7x - 3.$$ Given that \(x = 3\) is a solution of the equation f\((x) = 0\), solve f\((x) = 0\) completely. [5]

1. Show that \(x = 4\) is a root of \(x^3 - 12x - 16 = 0\). [2]
2. Hence completely factorise the expression \(x^3 - 12x - 16\). [3]

Given that \(x = -\frac{1}{2}\) is the real solution of the equation $$2x^3 - 11x^2 + 14x + 10 = 0,$$ find the two complex solutions of this equation. [6]

$$f(x) = 4x^3 - 12x^2 + 2x - 6$$

1. Use the factor theorem to show that \((x - 3)\) is a factor of \(f(x)\). [2]
2. Hence show that \(3\) is the only real root of the equation \(f(x) = 0\) [4]

7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

6. $$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$

1. 1. Calculate f(2)
   2. Write \(\mathrm { f } ( x )\) as a product of two algebraic factors. Using the answer to (a)(ii),
2. prove that there are exactly two real solutions to the equation $$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
3. deduce the number of real solutions, for \(7 \pi \leqslant \theta < 10 \pi\), to the equation $$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$

One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]

\(p(x) = x^3 - 5x^2 + 3x + a\), where \(a\) is a constant. Given that \(x - 3\) is a factor of \(p(x)\), find the value of \(a\) Circle your answer. [1 mark] \(-9\) \quad\quad \(-3\) \quad\quad \(3\) \quad\quad \(9\)

1

1. The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - a x - 12\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\).
2. When \(a\) has this value, find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).

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

1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
2. Find the value of \(a\) and the value of \(b\).

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ), the remainder is 7

1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
2. Find the value of \(a\) and the value of \(b\)

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7. $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$ Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that \(2 a + b = - 7\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 36
2. find the value of \(a\) and the value of \(b\). \(\mathrm { f } ( x )\) can be written in the form $$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$
3. 1. Find \(\mathrm { Q } ( x )\).
   2. Prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real root. You must justify your answer and show all your working.

4 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + k x + c\). The value of \(\mathrm { f } ( 0 )\) is 6, and \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the values of \(k\) and \(c\).

2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 9 x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) and \(( 2 x + 1 )\) are factors of \(\mathrm { p } ( x )\).
Find the values of \(a\) and \(b\).

5

1. Given that ( \(x + 2\) ) and ( \(x + 3\) ) are factors of $$5 x ^ { 3 } + a x ^ { 2 } + b$$ find the values of the constants \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise $$5 x ^ { 3 } + a x ^ { 2 } + b$$ completely, and hence solve the equation $$5 ^ { 3 y + 1 } + a \times 5 ^ { 2 y } + b = 0$$ giving any answers correct to 3 significant figures.

9

1. The polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 3 x + 3$$ Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\), and hence find the other two roots.
2. Hence solve the equation $$\tan ^ { 3 } x - \tan ^ { 2 } x - 3 \tan x + 3 = 0$$ for \(0 \leqslant x \leqslant 2 \pi\). Give each solution for \(x\) in an exact form.

The polynomial \(p(x)\) is defined by $$p(x) = ax^3 + 3x^2 + bx + 12,$$ where \(a\) and \(b\) are constants. It is given that \((x + 3)\) is a factor of \(p(x)\). It is also given that the remainder is 18 when \(p(x)\) is divided by \((x + 2)\).

1. Find the values of \(a\) and \(b\). [5]
2. When \(a\) and \(b\) have these values,
   1. show that the equation \(p(x) = 0\) has exactly one real root, [4]
   2. solve the equation \(p(\sec y) = 0\) for \(-180° < y < 180°\). [3]

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + 11 x ^ { 2 } + a x + a$$ where \(a\) is a constant. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).

1. Use the factor theorem to show that \(a = - 4\).
2. When \(a = - 4\),
   1. factorise \(\mathrm { p } ( x )\) completely,
   2. solve the equation \(6 \sec ^ { 3 } \theta + 11 \sec ^ { 2 } \theta + a \sec \theta + a = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]

1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]

The polynomial f(x) is defined by \(f(x) = x^3 - 9x^2 + 7x + 33\).

1. Find the remainder when f(x) is divided by \((x + 2)\). [2]
2. Show that \((x - 3)\) is a factor of f(x). [1]
3. Solve the equation f(x) = 0, giving each root in an exact form as simply as possible. [6]

Factorise completely $$x^3 - 4x^2 + 3x.$$ [3]

Factorise fully \(25 x - 9 x ^ { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-02_37_42_2700_1909}

4. The polynomial \(P ( x )\) is defined as follows: \(P ( x ) \equiv x ^ { 8 } + 8 x ^ { 7 } + 28 x ^ { 6 } + 56 x ^ { 5 } + 70 x ^ { 4 } + 56 x ^ { 3 } + 28 x ^ { 2 } + 8 x , x \in \mathbb { R }\) By first factorising \(P ( x )\) find all of its real roots.
[0pt]

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 \(k\).

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 \(b\).

6 It is given that \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).

1. Show that \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2 = 0\).
2. By first using the factor theorem, factorise \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2\) completely.
3. Hence solve the equation \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).

6 It is given that \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).

1. Show that \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2 = 0\).
2. By first using the factor theorem, factorise \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2\) completely.
3. Hence solve the equation \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).

7

1. Use the factor theorem to show that ( \(2 x + 3\) ) is a factor of $$8 x ^ { 3 } + 4 x ^ { 2 } - 10 x + 3$$
2. Show that the equation \(2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }\) can be expressed as $$8 \cos ^ { 3 } \theta + 4 \cos ^ { 2 } \theta - 10 \cos \theta + 3 = 0 .$$
3. Solve the equation \(2 \cos 2 \theta = \frac { 6 \cos \theta - 5 } { 2 \cos \theta + 1 }\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$

1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
2. 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.

4. Let \(f ( x )\) be given by: $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$ a) Use the factor theorem to show that ( \(x + 3\) ) is a factor of \(f ( x )\) b) Factorise \(f ( x )\) into a linear and a quadratic factor and hence find exact values for all of the solutions of the equation \(f ( x ) = 0\), showing detailed reasoning with your working
c) Hence write down the one solution to the equation $$e ^ { 3 x } + e ^ { 2 x } - 12 e ^ { x } - 18 = 0$$ in the form \(\ln ( a + \sqrt { b } )\) [0pt]

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$ where \(a\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
3. Hence find the exact values of \(y\) that satisfy the equation \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6\), where \(k\) is a constant. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 42 . Use the remainder theorem to find the value of \(k\). Hence find a root of \(\mathrm { f } ( x ) = 0\).

1. \(f ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).

Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is a remainder of - 35 ,

1. find the value of the constant \(k\),
2. find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 3 x - 2 )\).

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 3 x ^ { 2 } - 5 x + a + 4 ,$$ where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(\mathrm { p } ( x )\) completely,
   2. find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

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

7 Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 4 x + 4\).
Hence solve the equation \(x ^ { 3 } - x ^ { 2 } - 4 x + 4 = 0\).

Find the remainder when \(f(x) = 4x^3 + 3x^2 - 2x - 6\) is divided by \((2x + 1)\). [3]

Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]

5. (a) Find, using algebra, all solutions of $$20 x ^ { 3 } - 50 x ^ { 2 } - 30 x = 0$$ (b) Hence find all real solutions of $$20 ( y + 3 ) ^ { \frac { 3 } { 2 } } - 50 ( y + 3 ) - 30 ( y + 3 ) ^ { \frac { 1 } { 2 } } = 0$$

5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .

1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).

Factorise and hence simplify \(\frac{3x^2 - 7x + 4}{x^2 - 1}\). [3]

Use an algebraic method to solve the equation \(12x^3 - 29x^2 + 7x + 6 = 0\). Show all your working. [6]

You are given that \(\text{f}(x) = x^3 + 9x^2 + 20x + 12\).

1. Show that \(x = -2\) is a root of \(\text{f}(x) = 0\). [2]
2. Divide \(\text{f}(x)\) by \(x + 6\). [2]
3. Express \(\text{f}(x)\) in fully factorised form. [2]
4. Sketch the graph of \(y = \text{f}(x)\). [3]
5. Solve the equation \(\text{f}(x) = 12\). [3]

3

1. Find the value of \(a\) for which ( \(x - 2\) ) is a factor of \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8\).
2. Show that, for this value of \(a\), the cubic equation \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8 = 0\) has only one real root.

2 The cubic polynomial \(3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8\) is denoted by \(\mathrm { f } ( x )\).

1. Given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { f } ( x )\) completely.

Express \(3x^3 + 5x^2 - 27x + 10\) in the form \((x - 2)(ax^2 + bx + c)\), where \(a\), \(b\) and \(c\) are integers. [3 marks]

10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$

1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
3. Find, using algebraic integration, the exact value of the area of \(R\).

\(x^2 + bx + c\) and \(x^2 + dx + e\) have a common factor \((x + 2)\) Show that \(2(d - b) = e - c\) Fully justify your answer. [4 marks]

2. $$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 22
Find the value of \(a\) and the value of \(b\).

$$f(x) = x^3 + kx - 20.$$ Given that f(x) is exactly divisible by \((x + 1)\),

1. find the value of the constant \(k\), [2]
2. solve the equation \(f(x) = 0\). [4]

When \(x^3 + 3x + k\) is divided by \(x - 1\), the remainder is 6. Find the value of \(k\). [3]

5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
2. 1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as a product of linear factors.
3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\), stating the values of \(x\) where the curve meets the \(x\)-axis.

You are given that • the coefficient of \(x^3\) in the expansion of \((5 + 2x^2)(x^3 + kx + m)\) is 29, • when \(x^3 + kx + m\) is divided by \((x - 3)\), the remainder is 59. Find the values of \(k\) and \(m\). [5]

1 Show that ( \(x - 2\) ) is a factor of \(3 x ^ { 3 } - 8 x ^ { 2 } + 3 x + 2\).

6

1. Use the factor theorem to show that ( \(x + 2\) ) is a factor of the expression $$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$ and hence factorise the expression completely.
2. Deduce the roots of the equation $$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$

12

1. Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\) 12
2. Factorise \(\mathrm { p } ( x )\) completely.
   12
3. Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$

\(f(x) = 2x^3 - x^2 + px + 6\), where \(p\) is a constant. Given that \((x - 1)\) is a factor of \(f(x)\), find

1. the value of \(p\), [2]
2. the remainder when \(f(x)\) is divided by \((2x + 1)\). [2]

Let \(f(x) = 8x^3 + 54x^2 - 17x - 21\).

1. Show that \(x + 7\) is a factor of \(f(x)\). [1]
2. Find the quotient when \(f(x)\) is divided by \(x + 7\). [2]

1 When the polynomial $$a x ^ { 3 } + 4 a x ^ { 2 } - 7 x - 5$$ is divided by \(( x + 2 )\), the remainder is 33 .
Find the value of the constant \(a\).

6. $$f ( x ) = - 6 x ^ { 3 } - 7 x ^ { 2 } + 40 x + 21$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\)
2. Factorise f(x) completely.
3. Hence solve the equation $$6 \left( 2 ^ { 3 y } \right) + 7 \left( 2 ^ { 2 y } \right) = 40 \left( 2 ^ { y } \right) + 21$$ giving your answer to 2 decimal places.

1. Given that \((x + 2)\) is a factor of $$4x^3 + ax^2 - (a + 1)x - 18,$$ find the value of the constant \(a\). [3]
2. When \(a\) has this value, factorise \(4x^3 + ax^2 - (a + 1)x - 18\) completely. [3]

4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 3\) ).
2. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
3. Factorise \(\mathrm { f } ( x )\) completely.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
2. Hence
   1. find the value of \(a\)
   2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$

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

1. Find the value of \(a\).
2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\).

$$f(x) = 2x^3 - 5x^2 + x + 2.$$

1. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
2. Fully factorise \(f(x)\). [4]
3. Solve the equation \(f(x) = 0\). [1]
4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2\pi\) for which $$2\sin^3 \theta - 5\sin^2 \theta + \sin \theta + 2 = 0,$$ giving your answers in terms of \(\pi\). [4]

1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + ( k + 1 ) x ^ { 2 } - m x + 3 k$$ where \(k\) and \(m\) are constants. Given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), express \(m\) in terms of \(k\).