$\newcommand{\D}{\displaystyle} \newcommand{\qheading}[4]{( \text{CIE, #1, #2, paper #3, question #4})} $
1. (CIE 2012 summer paper 11 question 2)
The expression $\D 2x^3+ax^2+bx-30$ is divisible by $\D x+2$ and leaves the remainder of -35 when divided by $\D 2x-1$. Find the values of the constants $\D a$ and $\D b$.[5]
2. (CIE, 2012, summer, paper 22, question 5)
It is given that $\D x-2$ is a factor of $\D f(x)=x^3+kx^2-8x-8.$
(i) Find the value of the integer $\D k$.
(ii) Using your of $\D k$, find the non-integer roots of the equation $\D f(x)=0$ in the form $\D a\pm \sqrt{b},$ where $\D a$ and $\D b$ are integers. [7]
3. (CIE 2012, winter, paper 12, question 10)
A function $\D f$ is such that $\D f(x)=4x^3+4x^2+ax+b.$ It is given that $\D 2x-1$ is a factor of both $\D f(x)$ and $\D f'(x).$
(i) Show that $\D b = 2$ and find the value of $\D a.$ [5]
Using the values of a and b from part (i),
(ii) find the remainder when $\D f(x)$ is divided by $\D x + 3,$ [2]
(iii) express $\D f(x)$ in the form $\D f(x) = (2x - 1)(px^2 + qx + r),$ where p, q and r are integers to be found, [2] (iv) find the values of x for which f(x) = 0. [2]
4. (CIE 2012, winter, paper 13, question 8)
(i) Given that $\D 3x^3 + 5x^2 + px + 8 = (x - 2)(ax^2 + bx + c),$ find the value of each of the integers a, b, c and p. [5]
(ii) Using the values found in part (i), factorise completely $\D 3x^3 + 5x^2 + px + 8.$ [2]
5. (CIE 2012, winter,paper 21, question 10 )
(i) The remainder when the expression $\D x^3 + 9x^2 + bx + c$ is divided by $\D x - 2$ is twice the remainder when the expression is divided by $\D x - 1.$ Show that c = 24. [5]
(ii) Given that $\D x + 8$ is a factor of $x^3 + 9x^2 + bx + 24,$ show that the equation $\D x^3 + 9x^2 + bx + 24 = 0$ has only one real root. [4]
6. (CIE 2013, summer, paper 12, question 7)
It is given that $\D f(x) = 6x^3 - 5x^2 + ax + b$ has a factor of $\D x + 2$ and leaves a remainder of 27 when divided by $\D x - 1.$
(i) Show that b = 40 and find the value of a. [4]
(ii) Show that $\D f(x) = (x + 2)(px^2 + qx + r),$ where p, q and r are integers to be found. [2]
(iii) Hence solve $\D f(x) = 0.$ [2]
7. (CIE 2013, summer, paper 21, question 12)
The function $\D f(x) =x^3+ x^2+ ax+ b$ is divisible by $\D x - 3$ and leaves a remainder of 20 when divided by $\D x + 1.$
(i) Show that b = 6 and find the value of a. [4]
(ii) Using your value of a and taking b as 6, find the non-integer roots of the equation $\D f(x) = 0$ in the form $p \pm \sqrt{q},$ where p and q are integers. [5]
8. (CIE 2013, winter, paper 11, question 6)
The function $\D f(x)= ax^3+ 4x^2+ bx -2,$ where a and b are constants, is such that $\D 2x - 1$ is a factor. Given that the remainder when $\D f (x)$ is divided by $\D x - 2$ is twice the remainder when $\D f (x)$ is divided by $\D x + 1,$ find the value of a and of b. [6]
$9(\mathrm{CIE} 2013, \mathrm{w}$, paper 13, question 1$)$
The coefficient of $x^{2}$ in the expansion of $(2+p x)^{6}$ is 60 .
(i) Find the value of the positive constant $p$.
(ii) Using your value of $p$, find the coefficient of $x^{2}$ in the expansion of $(3-x)(2+p x)^{6}$.
10.$\qheading{2014}{summer }{13}{3}$
(i) Find, in terms of p, the remainder when $\D x^3 + px^2 + p^2x + 21$ is divided by $\D x + 3.$ [2]
(ii) Hence find the set of values of p for which this remainder is negative. [3]
11. $\qheading{2014}{summer}{21}{4}$
The expression $\D 2x^3+ ax^2+ bx+ 12$ has a factor $x - 4$ and leaves a remainder of -12 when divided by $\D x - 1.$ Find the value of each of the constants a and b. [5]
12. $\qheading{2014}{summer}{22}{3}$
(i) Given that $\D x + 1$ is a factor of $\D 3x^3- 14x^2- 7x+ d$, show that d = 10. [1]
(ii) Show that $\D 3x^3- 14x^2- 7x+ 10$ can be written in the form $\D (x+1)(ax^2+bx+c)$ , where a, b and c are constants to be found. [2]
(iii) Hence solve the equation $\D 3x^3- 14x^2- 7x+ 10= 0$. [2]
13 (CIE $2014, \mathrm{w}$, paper 23, question 1$)$
The expression $\mathrm{f}(x)=3 x^{3}+8 x^{2}-33 x+p \quad$ has a factor of $x-2$.
(i) Show that $p=10$ and express $\mathrm{f}(x)$ as a product of a linear factor and a quadratic factor. $[4]$
(ii) Hence solve the equation $\mathrm{f}(x)=0$. [2]
14 (CIE $2015, \mathrm{~s}$, paper 11, question 6$)$
The polynomial $\mathrm{f}(x)=a x^{3}-15 x^{2}+b x-2 \quad$ has a factor of $2 x-1 \quad$ and a remainder of 5 when divided by $\quad x-1$.
(i) Show that $b=8$ and find the value of $a$. [4]
(ii) Using the values of $a$ and $b$ from part (i), express $\mathrm{f}(x)$ in the form $(2 x-1) \mathrm{g}(x)$, where $\mathrm{g}(x)$ is a quadratic factor to be found.
(iii) Show that the equation $\mathrm{f}(x)=0$ has only one real root. $[2]$
15 (CIE 2015, s, paper 22, question 12)
(i) Show that $x=-2$ is a root of the polynomial equation $15 x^{3}+26 x^{2}-11 x-6=0$. $[1]$
(ii) Find the remainder when $15 x^{3}+26 x^{2}-11 x-6 \quad$ is divided by $x-3$. $[2]$
(iii) Find the value of $p$ and of $q$ such that $15 x^{3}+26 x^{2}-11 x-6 \quad$ is a factor of $15 x^{4}+p x^{3}-37 x^{2}+q x+6$
16 (CIE 2015, w, paper 21, question 1)
It is given that $\mathrm{f}(x)=4 x^{3}-4 x^{2}-15 x+18$.
(i) Show that $x+2$ is a factor of $f(x)$. $[1]$
(ii) Hence factorise $\mathrm{f}(x)$ completely and solve the equation $\mathrm{f}(x)=0$. $[4]$
17 (CIE 2015, w, paper 23, question 1)
Find the equation of the tangent to the curve $y=x^{3}+3 x^{2}-5 x-7 \quad$ at the point where $x=2 . \quad[5]$
18 (CIE 2015, w, paper 23, question 5)
The roots of the equation $x^{3}+a x^{2}+b x+c=0 \quad$ are 1,3 and 3 . Show that $c=-9$ and find the value of $a$ and of $b$.
19 (CIE 2016, march, paper 12 , question 7)
The polynomial $\mathrm{f}(x)=a x^{3}+7 x^{2}-9 x+b$ is divisible by $2 x-1$. The remainder when $\mathrm{f}(x)$ is divided by $x-2$ is 5 times the remainder when $\mathrm{f}(x)$ is divided by $x+1$.
(i) Show that $a=6$ and find the value of $b$. [4]
(ii) Using the values from part (i), show that $\mathrm{f}(x)=(2 x-1)\left(c x^{2}+d x+e\right)$, where $c, d$ and $e$ are integers to be found.
(iii) Hence factorise $\mathrm{f}(x)$ completely.
20 (CIE 2016, s, paper 11, question 10)
(i) Given that $\mathrm{f}(x)=4 x^{3}+k x+p$ is exactly divisible by $x+2$ and $\mathrm{f}^{\prime}(x)$ is exactly divisible by $2 x-1$, find the value of $k$ and of $p$.
(ii) Using the values of $k$ and $p$ found in part (i), show that $f(x)=(x+2)\left(a x^{2}+b x+c\right)$, where $a, b$ and $c$ are integers to be found.
(iii) Hence show that $\mathrm{f}(x)=0$ has only one solution and state this solution. $[2]$
21 (CIE 2016, s, paper 22, question 4) Do not use a calculator in this question.
The polynomial $p(x)=2 x^{3}-3 x^{2}+q x+56$ has a factor $x-2$
(i) Show that $q=-30$. $[1]$
(ii) Factorise $\mathrm{p}(x)$ completely and hence state all the solutions of $\mathrm{p}(x)=0$. $[4]$
22 CIE 2016, w, paper 11, question 9)
Do not use a calculator in this question.
The polynomial $p(x)$ is $a x^{3}-4 x^{2}+b x+18$. It is given that $p(x)$ and $p^{\prime}(x)$ are both divisible by $2 x-3$
(i) Show that $a=4$ and find the value of $b$. $[4]$
(ii) Using the values of $a$ and $b$ from part (i), factorise $p(x)$ completely. $[2]$
(iii) Hence find the values of $x$ for which $p(x)=x+2$. $[3]$
23 (CIE 2016, w, paper 23, question 4)
The cubic given by $p(x)=x^{3}+a x^{2}+b x-24$ is divisible by $x-2$. When $p(x)$ is divided by $x-1$ the remainder is $-20$.
(i) Form a pair of equations in $a$ and $b$ and solve them to find the value of $a$ and of $b$. [4]
(ii) Factorise $\mathrm{p}(x)$ completely and hence solve $\mathrm{p}(x)=0$. $[4]$
24 (CIE 2017, march, paper 22, question 3)
The polynomial $\mathrm{p}(x)$ is $x^{4}-2 x^{3}-3 x^{2}+8 x-4$
(i) Show that $p(x)$ can be written as $(x-1)\left(x^{3}-x^{2}-4 x+4\right)$. $[1]$
(ii) Hence write $\mathrm{p}(x)$ as a product of its linear factors, showing all your working. $[4]$
25 (CIE 2017, s, paper 11, question 2)
It is given that $p(x)=x^{3}+a x^{2}+b x-48$. When $p(x)$ is divided by $x-3$ the remainder is 6 . Given that $\mathrm{p}^{\prime}(1)=0$, find the value of $a$ and of $b$.
26 (CIE 2017, s, paper 13 , question 8) It is given that $p(x)=2 x^{3}+a x^{2}+4 x+b$, where $a$ and $b$ are constants. It is given also that $2 x+1$ is a factor of $\mathrm{p}(x)$ and that when $\mathrm{p}(x)$ is divided by $x-1$ there is a remainder of $-12$.
(i) Find the value of $a$ and of $b$. $[5]$
(ii) Using your values of $a$ and $b$, write $\mathrm{p}(x)$ in the form $(2 x+1) q(x)$, where $\mathrm{q}(x)$ is a quadratic expression.
(iii) Hence find the exact solutions of the equation $p(x)=0$.
27 (CIE 2017, s, paper 22, question 3)
Without using a calculator, factorise the expression $10 x^{3}-21 x^{2}+4 .$
28 (CIE 2017, w, paper 11, question 2) The polynomial $\mathrm{p}(x)$ is $a x^{3}+b x^{2}-13 x+4$, where $a$ and $b$ are integers. Given that $2 x-1$ is a factor of $\mathrm{p}(x)$ and also a factor of $\mathrm{p}^{\prime}(x)$
(i) find the value of $a$ and of $b$. $[5]$
Using your values of $a$ and $b$,
(ii) find the remainder when $p(x)$ is divided by $x+1$.
29 CIE 2017, w, paper 12, question 7) A polynomial $p(x)$ is $a x^{3}+8 x^{2}+b x+5$, where $a$ and $b$ are integers. It is given that $2 x-1$ is a factor of $\mathrm{p}(x)$ and that a remainder of $-25$ is obtained when $\mathrm{p}(x)$ is divided by $x+2$.
(i) Find the value of $a$ and of $b$. $[5]$
(ii) Using your values of $a$ and $b$, find the exact solutions of $p(x)=5$.
30 (CIE 2017, w, paper 23, question 11)
The cubic equation $x^{3}+a x^{2}+b x-36=0$ has a repeated positive integer root.
(i) If the repeated root is $x=3$ find the other positive root and the value of $a$ and of $b$. $[4]$
(ii) There are other possible values of $a$ and $b$ for which the cubic equation has a repeated positive integer root. In each case state all three integer roots of the equation.
31 (CIE 2018, march, paper 12 , question 1)
The remainder obtained when the polynomial $p(x)=x^{3}+a x^{2}-3 x+b \quad$ is divided by $x+3$ is twice the remainder obtained when $p(x)$ is divided by $x-2$. Given also that $p(x)$ is divisible by $x+1$, find the value of $a$ and of $b$.
32 (CIE 2018, s, paper 21, question 4)
Do not use a calculator in this question.
It is given that $x+4$ is a factor of $p(x)=2 x^{3}+3 x^{2}+a x-12 .$ When $p(x)$ is divided by $x-1$ the remainder is $b$.
(i) Show that $a=-23$ and find the value of the constant $b$. [2]
(ii) Factorise $p(x)$ completely and hence state all the solutions of $p(x)=0$. [4]
It is given that $x+3$ is a factor of the polynomial $p(x)=2 x^{3}+a x^{2}-24 x+b$. The remainder when $\mathrm{p}(x)$ is divided by $x-2$ is $-15$. Find the remainder when $\mathrm{p}(x)$ is divided by $x+1$.
Answers:
$\D
1. $a = 5; b = -13$
2. (i) $k = 4$
(ii) $-3\pm \sqrt{5}$
3. (i) $a = -7$
(ii)$-49 $
(iii) $(2x - 1)(2x^2 + 3x - 2)$
(iv)$x = 0:5;-2$
4. (i) $p = -26; a = 3; b = 11; c = -4 $
(ii)$(x - 2)(x + 4)(3x - 1)$
5. (ii) $x^2 + x + 3 = 0 : b^2 - 4ac < 0$
6. (i) $a = -14; b = 40 $
(ii)$f(x) = (x + 2)(6x^2 - 17x + 20)$
(iii)$x = -2$
7. (i) $a = -14; b = 6$
(ii)$-2\pm \sqrt{6}$
8. $a = -2; b = 2.5$
9. (i) $p = 0.5(ii)84$
10. (i) $9p - 3p^2 - 6 $
(ii) $p < 1; q > 2$
11. $a = -3; b = -23$
12. (ii)$3x^2 - 17x + 10 $
(iii)$-1; 5; 2/3 $
13. (i) $(x-2)(3 x-1)(x+5)$
(ii) $x=2,-5,1 / 3$
14. (i) $a=14, b=8$
(ii) $(2 x-1)\left(7 x^{2}-4 x+2\right)$
15. (ii) $600($ iii $) p=11, q=5$
16. (ii) $(x+2)(2 x-3)(2 x-3)$
$x=-2,1.3$
17. $y=19 x-35$
18. $a=-7, b=15, c=-9$
19. (i) $a=6, b=2$
(ii) $(2 x-1)\left(3 x^{2}+5 x-2\right)$
(iii) $(2 x-1)(3 x-1)(x+2)$
20. (i) $k=-3, p=26$
(ii) $(x+2)\left(4 x^{2}-8 x+13\right)$
(iii) $x=-2$
21. (ii) $x=2,-4,3.5$
22. (i) $a=4, b=-15$
(ii) $(x+2)(2 x-3)^{2}$
(iii) $x=-2,1,2$
23. (i) $a=5, b=-2$
(ii) $x=2,-3,-4$
24. $(x-1)^{2}(x+2)(x-2)$
25. $a=12, b=-27$
26. (i) $a=-27, b=9$
(ii) $(2 x+1)\left(x^{2}-14 x+9\right)$
(iii) $x=7 \pm 2 \sqrt{10},-1 / 2$
27. $(x-2)(2 x-1)(5 x+2)$
28. (i) $a=12, b=4$
(ii) 9
29. (i) $a=12, b=-17$
(ii) $x=0,-1 / 3 \pm \sqrt{55} / 6$
30. (i) $4, a=-10, b=33$
(ii) $x=6,6,1$
$x=2,2,9$
$x=1,1,36$
31. $a=10, b=-12$
32. (i) $a=-23, b=-30$
(ii) $-1 / 2,3,-4$
33. $a=-7, b=45,60$
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