$\def\frac{\dfrac}$

**Group (2015-2019)**

**1. (2015/Myanmar /q2or )**

(OR) Given that the expression $2 x^{3}+a x^{2}+b x+\mathrm{c}$ leaves the same remainder when divided by $x-2$ or by $x+1$, prove that $a+b=-6$. (3 marks)

**2. (2015/Myanmar /q8a )**

The expression $x^{3}+a x^{2}+b x+3$ is exactly divisible by $x+3$ but it leaves a remainder of 91 when divided by $x-4$. What is the remainder when it is divided by $x+2 ?$ (5 marks)

**3. (2015/FC /q2or )**

(OR) Given that the expression $x^{3}-p x^{2}+q x+\dot{r}$ leaves the same remainder when divided $s y x+1$ or $x-2$. Find $p$ in terms of $q$ (3 marks)

**4. (2015/FC /q8a )**

Given that $f(x)=x^{3}+p x^{2}-2 x+4 \sqrt{3}$ has a factor $x+\sqrt{2}$, find the value of $p$. Show that $x-2 \sqrt{3}$ is also a factor and solve the equation $f(x)=0$. (5 marks)

**5. (2016/Myanmar /q2or )**

(OR) Find the value of $k$ if $4 x^{7}+5 x^{3}-2 k x^{2}+7 k-4$ has a remainder of 12 when divided by $x+1$

**6. (2016/Myanmar /q8a )**

The cubic polynomial $f(x)$ is such that the coefficient of $x^{3}$ is $-1$ and the roots of the equation $f(x)=0$ are 1,2 and $k$. Given that $f(x)$ has a remainder of 8 when divided by $x-3$, find the value of $k$ and the remainder when $f(x)$ is divided by $x+3$

**7. (2016/FC /q2or )**

(OR) Given that $x^{3}-2 x^{2}-3 x-11$ and $x^{3}-x^{2}-9$ have the same remainder when divided by $x+a$, determine the values of $a$.

**8. (2016/FC /q8a )**

The expression $a x^{3}-(a+3 b) x^{2}+2 b x+c$ is exactly divisible by $x^{2}-2 x$. When the expression is divided by $x-1$, the remainder is 8 more than when it is divided by $x+1$. Find the values of $a, b$ and $c$, hence factorize the expression completely.

**9. (2017/Myanmar /q2or )**

(OR) If $x^{3}+p x^{2}-8 q x+5$ and $2 x^{3}-q x^{2}+4 p x-18$ have a common factor $x-2$ find the values of $p$ and $q$.

Q2(OR) Solution

**10. (2017/Myanmar /q8a )**

If $x+2$ is a factor of $x^{3}-a x-6$, then find the remainder when $2 x^{3}+a x^{2}-6 x+9$ is divided by $x+1$

Q8(a) Solution

**11. (2017/FC /q2or )**

(OR) $x^{2}-1$ is a factor of $x^{3}+a x^{2}-x+b$. When the expression is divided by $x-2$ the remainder is 15 . Find the values of a and b. $(3$ marks)

**12. (2017/FC /q8a )**

Given that $4 \mathrm{x}^{4}-9 \mathrm{a}^{2} \mathrm{x}^{2}+2\left(\mathrm{a}^{2}-7\right) \mathrm{x}-18$ is exactly divisible by $2 \mathrm{x}-3 \mathrm{a}$, show that $\begin{array}{ll}a^{3}-7 a-6=0 \text { and hence find the possible values of a. } & \text { (5 marks) }\end{array}$

**13. (2018/Myanmar /q2or )**

(OR) The remainder when $2 x^{3}+k x^{2}+7$ is divided by $x-2$ is half the remainder when the same expression is divided by $2 x-1$. Find the value of $k$.

Click for Solution

**14. (2018/Myanmar /q8a )**

The expression $p x^{3}-5 x^{2}+q x+10$ has factor $2 x-1$ but leaves a remainder of $-20$ when divided by $x+2$. Find the values of $p$ and $q$ and factorize the expression completely.

Click for Solution

**15. (2018/FC /q2or )**

(OR) The expression $x^{3}-2 x^{2}-k x+6$ and $x^{3}+x^{2}+(8-k) x+10$ have the same remainder when divided by $x+a$. Show that $3 a^{2}-8 a+4=0$

**16. (2018/FC /q8a )**

Given $f(x)=x^{3}+p x^{2}-2 x+4 \sqrt{3}$ has a factor $x-2 \sqrt{3}$, find the value of $p$. Show that $x+\sqrt{2}$ is also a factor and solve the equation $f(x)=0$.

**17. (2019/Myanmar /q1b )**

The expression $2 x^{2}+5 x-3$ leaves a remainder of $2 p^{2}-3 p$ when divided by $2 x-p$. Find the values of $\mathrm{p}$. $(3 \mathrm{marks})$ Click for Solution

**18. (2019/Myanmar /q6b )**

The expression $a x^{3}-x^{2}+b x-1$ leaves the remainders of $-33$ and 77 when divided by $\mathrm{x}+2$ and $\mathrm{x}-3$ respectively. Find the values of $\mathrm{a}$ and $\mathrm{b}$, and the remainder when divided by $\mathrm{x}-2$ (5 marks) Click for Solution

**19. (2019/FC /q1b )**

When $\mathrm{f}(\mathrm{x})=(\mathrm{x}-1)^{3}+6(\mathrm{px}+4)^{2}$ is divided by $\mathrm{x}+2$, the remainder is $-3$. Find the values of p. (3 marks) Click for Solution 1(b)

**20. (2019/FC /q6b )**

The polynomial $a x^{3}+b x^{2}-5 x+2 a$ is exactly divisible by $x^{2}-3 x-4$. Find the values of a and $b$. What is the remainder when it is divided by $x+2 ?$ (5 marks) Click for Solution 6(b)

### Answer (2015-2019)

1. Prove

2. 7

3. $p=q+3$

4. $p=-2\sqrt 3,x=2\sqrt 3,\pm \sqrt 2$

5. $ \quad k=5$

6. $k=7, R=200$

7. $a=1$ or 2

8. $a=2, b=1, c=0, f(x)=x(x-2)(2 x-1)$

9. $\quad p=\frac{3}{4}, \quad q=1$

10. 20

11. $a=3,b=-3$

12. $a=-1,3,-2$

13. $k=-5$

14. $p=6, q=-19, f(x)=(2 x-1)(3 x+5)(x-2)$

15. Show

16. $x=2 \sqrt{3}, \sqrt{2},-\sqrt{2}$

17. $p=\frac 23$ or $p=3$

18. $a=3,b=2,23$

19. 1 or 3

20. $a=2,b=-7, -30$

**Group (2014)**

1. When $(x+k)^{2014}+(2 x+1)^{3}$ is divided by $x+2$ the remainder is 10 find the values of $k$. $\qquad\mbox{ (3 marks)}$

2. Given that the expression $x^{3}+a x^{2}+b x+c$ leaves the same remainder when divided by $x-1$ or $x+2$, find $a$ in terms of $b$. $\qquad\mbox{ (3 marks)}$

3. The remainder when $x^{3}-6 x+p$ is divided by $x-3$ is twice the remainder when $x^{2}-4 x+p$ is divided by $x+5 .$ Find $p$. $\qquad\mbox{ (3 marks)}$

4. When the expression $7 x^{21}-5 x^{15}+a x^{6}$ is divided by $x+1$, the remainder is 2 . Find the value of $a$. Hence find the remainder when the expression is divided by $x-1$. $\qquad\mbox{ (3 marks)}$

5. If the expression $f(x)=k x^{2}+5 x-6$ is divisible by $2 x+3$, find the remainder when it is divided by $3 x-2$. $\qquad\mbox{ (3 marks)}$

6. Find what values $k$ must have in order that $(x-k)$ may be a factor of $4 x^{3}-(3 k+2) x^{2}-\left(k^{2}-1\right) x+3$. $\qquad\mbox{ (3 marks)}$

7. $x^{2}-1$ is a factor of $x^{3}+a x^{2}-x+b$. When the expression is divided by $x-2$, the remainder is 15 . Find the values of $a$ and $b$. $\qquad\mbox{ (3 marks)}$

8. Given $f(x)=2 x^{3}+\alpha x^{2}-7 a^{2} x-6 a^{3}$, determine whether or not $x-a$ and $x+a$ are factors of $f(x)$. Hence factorize $f(x)$ completely. $\qquad\mbox{ (3 marks)}$

9. If $(k x+1)$ is a common factor of the polynomials $2 x^{2}+7 x+3$ and $2 x^{2}-5 x-3$, find the value of $k$ and hence find also the remainder when $2 x^{3}+x^{2}-18 x-9$ is divided by $x+k$. $\qquad\mbox{ (3 marks)}$

10. Given that $f(x)=x^{2 n}-(p+1) x^{2}+p$ where $n$ and $p$ are positive integers. Show that $x-1$ is a factor of $f(x)$ for all values of $p$. When $p=4$ find the value of $n$ for which $x-2$ is a factor of $f(x)$. $\qquad\mbox{ (3 marks)}$

11. Given that $f(x)=x^{2 n}-(p+1) x^{2}+p$, where $n$ and $p$ are positive integers. Show that $x-1$ is a factor of $f(x)$, for all values of $p$. When $p=4$, find the value of $n$ for which $x-2$ is a factor of $f(x)$ and hence factorize $f(x)$ completely. $\qquad\mbox{ (5 marks)}$

12. The expression $x^{2 n}-k$ has $x+3$ as a factor and leaves a remainder of $-80$ when divided by $x+1 .$ Calculate the values of $n$ and $k$ where both are integers. With these values of $n$ and $k$, factorise $x^{2 n}-\mathrm{k}$ completely. $\qquad\mbox{ (5 marks)}$

13. The expression $\alpha x^{3}-6 a x+b$ leaves the remainders of 1 and 2 when divided by $x-1$ and $x-2$ respectively. Find the values of $a, b$ and hence find the remainder when the expression is divided by $x+4$. $\qquad\mbox{ (5 marks)}$

14. Given that the polynomial $x^{2}-10 x+14$ leaves the same remainder when divided by $x+2 b$ or $x+2 c$ where $3 b-2 c=0$ and $b \neq c$. Find the values of $b$ and $c$. $\qquad\mbox{ (5 marks)}$

15. The remainder when $x^{4}+3 x^{2}-2 x+2$ is divided by $x+a$ is the square of the remainder when $x^{2}-3$ is divided by $x+a .$ Find $a$. $\qquad\mbox{ (5 marks)}$

16. The expression $6 x^{3}+a x^{2}+b x+10$ has factor $2 x-1$ but leaves remainder $-20$ when divided by $x+2$. Find $a, b$ and factorise the expression completely. $\qquad\mbox{ (5 marks)}$

17. If $x+4$ is a factor of $f(x)=a(x+1)^{2}+b(x+1)+9 .$ The remainder when $f(x)$ is divided by $x+3$ is $-11$. Find the values of $a$ and $b$. Find also the solution set of the equation $f(x)=0$. $\qquad\mbox{ (5 marks)}$

18. Given that $x+5$ is a common factor of $x^{3}+p x^{2}-q x+15$ and $x^{3}-x^{2}-(q+5) x+40$. Find the values of $p$ and $q$. Hence factorize $x^{3}+p x^{2}-q x+15$ completely. $\qquad\mbox{ (5 marks)}$

19. Given that the equations $a x^{3}+4 x^{2}-5 x-10=0$ and $a x^{3}-9 x-2=0$ have a common root. What are the possible values of $a ?$ $\qquad\mbox{ (5 marks)}$

20. Given that $16 x^{4}-4 x^{3}-4 a^{2} x^{2}+7 a x+18$ is divisible by $2 x+a$, show that $a^{3}-7 a^{2}+36=0$ and find the possible values of $a$. $\qquad\mbox{ (5 marks)}$

21. Solve the equation $x^{4}-9 x^{2}=4 x-12$. $\qquad\mbox{ (5 marks)}$

**Answer (2014)**

1. $k=2 \pm \sqrt[2014]{37}$

2. $a=b+3 \quad$

3. $p=-81$

4. $a=4,6$

5. 0

6. $k=\frac{3}{2}$ or $-1$

7. $a=3, b=3$

8. $(x-a)$ is not a factor, $x+a$ is a factor, $(x+a),(2 x+3 a)$ and $(x-2 a)$

9. $k=2,-25$

10. $n=2$

11. $n=2;(x+2),(x-2),(x+1)$ and $(x-1)$

12. $k=81, n=2 ;(x-3)(x+3)\left(x^{2}+9\right)$

13. $a=1, b=6;-34$

14. $b=-2, c=-3$

15. $a=\frac{7}{9}($ or $)-1$

16. $a=-5, b=-19; (2 x-1),(3 x+5),(x-2)$

17. $a-7,b=24,\left\{-4,-\frac{10}{7}\right\}$

18. $p=1, q=17;(x-1),(x+5)$ and $(x-3)$

19. $a=2$ or 11

20. $a=3$ or 6 or $-2 \quad$

21. $x=1$ or $-2$ or 3

**Group (2013)**

1. When the expression $2 k x^{2}-k^{2} x-14$ is divided by $x-3$, the remainder is 10 Calculate the value of $k$. (3 marks)

2. When the polynomial $k^{2} x^{4}-k x^{2}-4, k>0$, is divided by $x+1$, the remainder is 16. Find the value of $k$ and the remainder when this polynomial is divided by $x-3$. (3 marks)

3. Given that $x^{3}-2 x^{2}-3 x-11$ and $x^{3}-x^{2}-9$ have the same remainder when divided by $x+a$, determine the values of $a$. (3 marks)

4. The expression $2 x^{2}+5 x-3$ leaves a remainder of $2 p^{2}-3 p$ when divided by $2 x-p$. Find the values of $p$. (3 marks)

5. The remainder when $2 x^{3}-5 x^{2}+3 x+k$ is divided by $x+1$ is equal to the remainder when $2 x^{3}+k x^{2}+1$ is divided by $2 x-1$, find the value of $k$. (3 marks)

6. Find the value of $a$ for which $(1-2 a) x^{2}+5 a x+(a-1)(a-8)$ is divisible by $x-2$ but not by $x-1$. (3 marks)

7. Given $f(x)=2 x^{3}+a x^{2}-7 a^{2} x-6 a^{3}$, determine whether or $\operatorname{not} x-a$ and $x+a$ are factors of $f(x)$. Hence, find the roots of $f(x)=0$ in terms of $a$. (5 marks)

8. If the equations $a x^{3}+4 x^{2}-5 x-10=0$ and $a x^{3}-9 x-2=0$ have a common root, find the values of $a$. (5 marks)

9. The expression $a x^{3}-x^{2}+b x-1$ leaves the remainders of $-33$ and 77 when divided by $x+2$ and $x-3$ respectively. Find the values of ' $a$ ' and ' $b$ ' and the remainder when divided by $x-2$. (5 marks)

10. The expression $a x^{2}+b x-1$ leaves remainder of $R$ when divided by $x+2$ and a remainder of $3 R+5$ when divided by $x-3$. Show that $a=3 b-1$. Given also that the expression is exactly divisible by $2 x-1$, evaluate $a$ and $b$. (5 marks)

11. If $f(x)=2 x^{4}+x^{3}-a x^{2}+b x+a+b-1$ has factors $x-2$ and $x+3$, find the constants $a$ and $b$. Hence factorize $f(x)$ completely. (5 marks)

12. Given that $2 x^{2}-x-1$ is a factor of $a x^{4}+x^{3}-b x^{2}+5 x+6$, find the values of $a$ and $b$. (5 marks)

13. Given that the equation $2 x^{3}+a x^{2}+b x-12=0$ has roots $x=1$ and $x=4$. Find the values of $a, b$ and the third root. (5 marks)

14. Solve $x^{4}+5 x^{3}+5 x^{2}-5 x-6=0$. (5 marks)

15. If $x-k$ is a factor of the expression $k x^{3}+5 x^{2}-7 k x-8$, where $k$ is a positive integer, find the numerical value of $k$. Hence find the other factors of the expression. (5 marks)

16. Show that the expression $x^{3}+(k+1) x^{2}+(k-6) x-6$ has a factor $x+1$ for all values of $k$. If the expression also has a factor $x+3$, find $k$ and the third factor. (5 marks)

17. Given that $f(x)=x^{2 n}-(p+1) x^{2}+p$ where $n$ and $p$ are positive integers, show that $x-1$ is a factor of $f(x)$ for all values of $p$. When $p=4$, find the value of $n$ for which $x-2$ is a factor of $f(x)$ and, for this case, factorize $f(x)$ completely. (5 marks)

18. The polynomials $f(x)=3 x^{3}+a x-7$ and $g(x)=2 x^{3}-3 x^{2}+2 b x-5$ have the same remainder when divided by $x-2$. Find ' $a$ ' in terms of $b$ '. (3 marks)

19. The expression $x^{2}+b x+a$ has the same remainder when divided by $x-b$ or $x-a$. Find the values of $a$ in terms of $b$. (3 marks)

20. When $a x^{2}+b x-6$ is divided by $x+3$, the remainder is $9 .$ In terms of a only, find the remainder when $2 x^{3}-b x^{2}+2 a x-4$ is divided by $x-2$. (3 marks)

21. Given that when $x^{2}+a x+b$ and $x^{2}+h x+k$ are divided by $x+p$, their remainder are equal. Express $p$ in terms of $a, b, h$ and $k$. (3 marks)

22. Given that $x-2 b$ is a factor of $a x^{2}-x-2 a$, find an expression for ' $a$ ' in terms of ' $b$ '. (3 marks)

**Answer (2013)**

1. $k=2$ (or) 4

2. $k=5 ; 1976$

3. $a=1$ (or) 2

4. $p=\frac{2}{3}$ (or) $3 \quad$

5. $k=15$

6. $a=4$

7. $(x-a)$ is not a factor and $(x+a)$ is a factor $x=-a$ (or) $-\frac{3 a}{2}$ (or) $2 a$

8. $a=2$ (or) 11

9. $a=3, b=2 ; 23$

10. $a=2,b=1$

11. $a=16$ or $b=3; (x-2),(x+3), (2x-3),(x+i)$

12. $a=2, b=14 \quad$

13. $a=-13, b=23 ;\frac{3}{2} \quad$

14. $x=-1$ (or) $-2$ (or) $-3$ (or) 1

15. $k=2 ;(2 x+1) \cdot(x+4) \quad$

16. $k=1 ;(x-2)$

17. $n=2 ;(x+2),(x-2),(x+1),(x-1)$

18. $a=2 b-9$

19. $a=b$ or $a=-2b$

20. $8(4-a)$

21. $p=\frac{k-b}{b-a}$

22. $a=\frac{b}{2 b^{2}-1}$

#### ** Group (2012)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | When $x^{3}+3 x^{2}-k x+4$ is divided by $x-2$, the remainder is $k$. Find the value of $k$. (3 marks) | |

2. | The sum of the remainders when $x^{3}+(k+8) x+k$ is divided by $x-2$ and $x+1$ is 0 . Find the value of $k$. (3 marks) | |

3. | $x^{3}-a x+a^{2}$ and $x^{3}+x^{2}-16$ have same remainder when divided by $x-a$. Find $a$. (3 marks) | |

4. | When the polynomial $6 x^{3}+p x^{2}-19 x+10$ is divisible by $2 x-1$, find the value of $p$. Find also the remainder when $f(x)$ is divided by $x+2$. (3 marks) | |

5. | Find the value of $p$ if $2 x^{3}+x^{2}+5 x+p$ is divisible by $(2 x-1)$. What is the remainder when the given polynomial is divided.by $(2 x+1)$. (3 marks) | |

6. | Given that $x+p$ is the common factor of $x^{2}+2 x-15$ and $2 x^{2}-5 x-3$, then find the value of $p$ and find the remainder when $3 x^{3}-4 x^{2}-5 x+6$ is divided by $x-p$. (3 marks) | |

7. | Given that $f(x)=2 x^{3}+x^{2}-24 x+q$ has a factor $x+2 \sqrt{3}$, find the value of $q$. Show that $x-2 \sqrt{3}$ is also a factor of $f(x)$ and find the other factor. (5 marks) | |

8. | The remainder when $a x^{3}+b x^{2}+2 x+3$ is divided by $x-1$ is twice the remainder when it is divided by $x+1$. Find the relation between $a$ and $b$. (3 marks) | |

9. | $x+2$ is a factor of $f(x)=a(x-1)^{2}+b(x-1)+a$. The remainder when $f(x)$ is - divided by $x+1$ is 5 . Find the value of $a$ and of $b$. (3 marks) | |

10. | Find all real roots of $1-x+x^{2}-x^{3}=0$. (3 marks) | |

11. | Show that $3 x-2$ is a factor of $6 x^{3}-x^{2}-20 x+12$ and find the other factors Hence solve the equation $6 x^{3}-x^{2}-20 x+12=0$. (5 marks) | |

12. | If $x+2$ is a factor of the expression $f(x)=a x^{3}+x^{2}-19 x+6$, find the value of $a$. and then solve the equation $f(x)=0$. (5 marks) | |

13. | The expression $2 x^{4}+a x^{3}+b x^{2}-3 x-4$ is exactly divisible by $x-4$ but it leaves a remainder of $-9$ when divided by $x-1$. What is remainder when it is divided by $x-2 ?$ (5 marks) | |

14. | The expression $p x^{3}-5 x^{2}+q x+10$ has factor $2 x-1$ but leaves a remainder of - 20 when divided by $x+2$. Find the values of $p$ and $q$ and factorize the expression completely. (5 marks) | |

15. | The expression $a x^{3}+b x^{2}-2 x^{2}+3$ has factor $x+3$ but leaves a remainder of 91 when divided by $x-4$. Find the values of $a$ and $b$. What is the remainder when it is divided by $x+2$. (5 marks) | |

16. | The polynomial $p x^{3}+q x^{2}-5 x-6$ is exactly divisible by $2 x^{2}+x-6 .$ Calculate the values of $p$ and $q$ and factorize the polynomial completely. (5 marks) | |

17. | The remainders when $f(x)=a x^{2}+b x+c$ is divided by $x-1, x+1, x-2$ are 1 , 25,1 respectively. Show that $f(x)$ is a perfect square. (5 marks) | |

18. | The expression $2 x^{3}+b x^{2}-c x+d$ leaves the same remainder when divided by $x-1$ or $x+2$ or $2 x-1$. Evaluate $b$ and $c$. Given also that the expression is exactly divisible by $x-2$, evaluate $d$. (5 marks) | |

19. | Find the value of $k$ for which $a-3 b$ is a factor of $a^{4}-7 a^{2} b^{2}+k b^{4}$. Hence for this value of $k$, factorise $a^{4}-7 a^{2} b^{2} \div k b^{4}$ completely. (5 marks) | |

20. | $f(x)=2 x^{3}+p x^{2}+q x-20$, where $p$ and $q$ are constants. Given that $x+2$ is a factor of $f(x)$ and that $x+2$ is also a factor $o$ ' $f^{\prime}(x)$, find the values of $p$ and $q$. (5 marks) | |

21. | The remainder when $x^{2 n}-7 x^{n}+5$ is divided by $(x-2)$ is 13, find the value of $n$. (3 marks) |

#### ** Answer (2012)**

$\quad\;\,$ | ||
---|---|---|

1. | $k=8$ | |

2. | $k=-5$ | |

3. | $a=\pm 4$ | |

4. | $p=-5 ;-20$ | |

5. | $p=-3 ;-5.5$ | |

6. | $p=-3 ;-96$ | |

7. | $q=-12 ; 2 x+1$ | |

8. | $3 a-b=-3$ | |

9. | $a=-3, b=-10$ | |

10. | $x=1$ | |

11. | $(2 x-3)$ and $(x+2) ;x=\frac{2}{3}$ (or) $x=\frac{3}{2}$ (or) $x=-2$ | |

12. | $a=6, x=-2$ (or) $x=\frac{1}{3}$ (or) $x=\frac{3}{2}$ | |

13. | $a=-9, b=5 ;$ the remainder $=-30$ | |

14. | $p=6, q=-19 ;(2 x-1),(3 x+5),(x-2)$ | |

15. | $a=1, b=2 ;$ remainder $=7$ | |

16. | $p=2, q=3 ;(x+2),(2 x-3),(x+1)$ | |

17. | Show | |

18. | $b=1,c=5,d=-10$ | |

19. | $k=-18,\left(a^{2}+2 b^{2}\right)(a+3 b)(a-3 b)$ | |

20. | $p=3,q=-12$ | |

21. | $n=3$ |

## ** Group (2011)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | When $(2 x-1)(4 x+p)(x+1)$ is divided by $2 x-3$, the remainder is $5 .$ Find $p$. (3 marks) | |

2. | The expressions $x^{2}+b x+a$ leaves the same remainder when divided by $x+2$ and $x-a$ where $a \neq-2$. Find the relation between $a$ and $b$. (3 marks) | |

3. | Given that the expression $x^{2}-5 x+7$ leaves the same remainder whether divided by $x-b$ or $x-c$, where $b \neq c$, show that $b+c=5$. (3 marks) | |

4. | The polynomials $f(x)=2 x^{3}+3 x-7$ and $g(x)=x^{3}-3 x^{2}+b x-5$ have the same remainder when divided by $x-2 .$ Find the value of $b$. (3 marks) | |

5. | The remainder when $2 x^{3}+k x^{2}+7$ is divided by $x-2$ is half the remainder when the same expression is divided by $2 x-1 .$ Find the value of $k$. (3 marks) | |

6. | If the expression $f(x)=6 x^{3}+13 x^{2}-40 x-4 p$ is divisible by $2 x-1$, find the value of $p$. Find also the remainder when $f(x)$ is divided by $x$. (3 marks) | |

7. | Determine whether or not $x+1$ is a factor of the polynomials $f(x)=3 x^{4}+x^{3}-x^{2}+3 x+2$ and $g(x)=x^{6}+2 x(x-1)+2$. (3 marks) | |

8. | If $(x-2)$ and $(x+3)$ are factors of the polynomial $f(x)=p x^{3}+x^{2}-13 x+p q$, find the value of $p$. (3 marks) | |

9. | Given that $x-p$ is the common factor of $x^{2}+3 x-10$ and $x^{2}+5 x-14$, find the value of $p$ and by using the value of $p$, show that $2 x^{3}+x^{2}-7 x-6$ is divisible by $ $x-p$. (3 marks) | |

10. | The expression $a x^{3}-x^{2}+b x-1$ leaves the remainders of $-33$ and 77 when divided by $x+2$ and $x-3$ respectively. Find the values of $a$ and $b$ and the remainder when divided by $x-2$. (5 marks) | |

11. | Show that $2 x+1$ is a factor of $6 x^{4}-5 x^{3}-10 x^{2}+5 x+4$ and find the other factors. (5 marks) | |

12. | Find the factors of $x^{4}-14 x^{3}+71 x^{2}-154 x+120$. (5 marks) | |

13. | Solve the equation $2 x^{3}+3 x^{2}=5 x+6$. (5 marks) | |

14. | The expression $a x^{3}+x^{2}+b x+6$ is exactly divisible by $x-2$ but it leaves a remainder of 43 when divided by $x-3 .$ Calculate the value of $a$ and $b$, and factorize the expression completely. (5 marks) | |

15. | Let $f(x)=x^{3}+a x^{2}+b x-22$. When $f(x)$ is divided by $x+3$, the remainder is $-28$. If $x+2$ is a factor of $f(x)$, find the values of $a$ and $b$. (5 marks) | |

16. | The expression $a x^{3}+b x^{2}-13 x-6$ has a factor $x-2$ but it leaves a remainder of $-4$ when divided by $x-1$. Calculate the values of $a$ and $b$, and what is the remainder when it is divided by $x+2 ?$ (5 marks) | |

17. | Find the values of $p$ and $q$ for which the expression $12 x^{4}+16 x^{3}+p x^{2}+q x-1$ is divisible by $4 x^{2}-1 .$ Hence, find the other factors of the expression. (5 marks) | |

18. | Given that $x+2$ is a common factor of $a x^{2}+(a+k) x+6$ and $(k-a) x^{2}+4 x-a$, then find the value of $a$ and $k$. (5 marks) | |

19. | If the equations $a x^{3}+4 x^{2}-5 x-10=0$ and $a x^{3}-9 x-2=0$ have a common root, find the values of $a$. (5 marks) |

#### ** Answer (2011)**

$\quad\;$ | $\,$ | |
---|---|---|

1. | $p=-5$ | |

2. | $a=2-b$ | |

3. | Prove | |

4. | $b=12$ | |

5. | $k=-5$ | |

6. | $p=-4 ; 16$ | |

7. | $(x+1)$ is a factor of $f(x)$ | |

8. | $p=2$ | |

9. | $p=2$ | |

10. | $a=3, b=2 ; 23$ | |

11. | $x+1, x-1,3 x-4$ | |

12. | $x-2, x-3, x-4, x-5$ | |

13. | $-2,-1, \frac{3}{2}$ | |

14. | $a=\frac{43}{15}, b=\frac{-247}{15} ; \frac{1}{15}(x-2)\left(43 x^{2}+101 x-45\right)$ | |

15. | $a=-8, b=-31$ | |

16. | $a=-7, b=22 ; 164$ | |

17. | $p=1, q=-4,3 x+1, x+1$ | |

18. | $a=4, k=7$ | |

19. | $a=2$ (or) 11 |

## ** Group (2010)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | When $(x+k)^{4}+(2 x+1)^{2}$ is divided by $x+2$ the remainder is 10, find the values of $k$.$\text{ (3 marks)}$ | |

2. | When the expression $x^{4}-k^{2} x^{3}-7 x^{2}+34 x-10 k$ is divided by $x+2$, the remainder is $-68$.Calculate the values of $k$.$\text{ (3 marks)}$ | |

3. | When the expression $2 k x^{4}-5 k^{2} x^{3}+23 x^{2}+5 x-6$ is divided by $x+2$, the remainder is 300.Calculate the values of $k$.$\text{ (3 marks)}$ | |

4. | When $f(x)=(x+2)^{3}(x-1)-k x+6$ is divided by $x+3$, the remainder is 28.Find value of $k$.Find also the remainder when $f(x)$ is divided by $x+2$.$\text{ (3 marks)}$ | |

5. | When the polynomial $k^{2} x^{16}+k, k<0$ is divided by $x+1$, the remainder is 12.Find the value of $k$.Find also the remainder when this polymomial is divided by $x-1$.$\text{ (3 marks)}$ | |

6. | The polynomials $f(x)=3 x^{3}+a x-7$ and $g(x)=2 x^{3}-3 x^{2}+2 b x-5$ have the same remainder when divided by $x-2.$ Find $a$ in terms of $b$.$\text{ (3 marks)}$ | |

7. | Find what value $p$ must have in order that $x+2$ ray be a factor of $2 x^{3}+p x^{2}-17 x+10$ and hence find the remainder when it is divided by $x-3$.$\text{ (3 marks)}$ | |

8. | If $x-k$ is a factor of $f(x)=x^{2}+(k-5) x-k^{2}+7 k-3$, find the values of $k$.$\text{ (3 marks)}$ | |

9. | The expressions $x^{3}-x^{2}-4 x+24$ and $x^{3}-7 x+6$ have the same remainder when divided by $x+k$.Find the possible values of $k$.$\text{ (5 marks)}$ | |

10. | Given that the polynomial $x^{2}-10 x+14$ leaves the same remainder when divided by $x+2 b$ or $x+2 c$, where $b-c=1.$ Find the values of $b$ and $c$.$\text{ (5 marks)}$ | |

11. | Given that the remainder when $f(x)=x^{3}-x^{2}+a x$ is divided by $x+a$ where $a>0$, is twice the remainder when $f(x)$ is divided by $x-2 a$, find the value of $a$.Find also the remainder when $f(x)$ is divided by $x-2$.$\text{ (5 marks)}$ | |

12. | Given that the expression $2 x^{3}+b x^{2}-c x+d$ leaves the same remainder when divided by $x+1$ or $x-2$ or $2 x-1$, find the values of $b$ and $c$.Given also that the expression is divisible by $x+2$, find the value of $d$.$\text{ (5 marks)}$ | |

13. | The polynomial $x^{3}-6 x^{2}+a x+b$ is divisible by $x^{2}-5 x+6.$ Find the value of $a$ and the value of $b$.$\text{ (5 marks)}$ | |

14. | The polynomial $x^{3}-6 x^{2}+p x+q$ is divisible by $x^{2}-5 x+6.$ Find the value of $p$ and the value of $q$.$\text{ (5 marks)}$ | |

15. | Solve $2 x^{3}+x^{2}-13 x+6=0$.$\text{ (5 marks)}$ | |

16. | Solve $3 x^{3}-4 x^{2}-17 x+6=0$.$\text{ (5 marks)}$ | |

17. | Solve $x^{4}+5 x^{3}+5 x^{2}-5 x-6=0$.$\text{ (5 marks)}$ | |

18. | The expression $x^{3}+a x^{2}+b x+c$ is divisible by both $x$ and $x-3$ and leaves a remainder of $-40$ when divided by $x+2$.Find the value of $a$, of $b$, of $c$, hence factorize the expression completely.$\text{ (5 marks)}$ | |

19. | Show that $3 x^{3}+(k-6) x^{2}+(k-5) x+4$ has a factor $x+1$ for all values of $k$.If the expression also has a factor $x-2$, find $k$ and the third factor.$\text{ (5 marks)}$ | |

20. | $x-3$ is a factor of the polynomial $x^{2}-(3 p+2) x+7 p+1$.Find the value of $p$ and the other factor of the polynomial.$\text{ (3 marks)}$ | |

21. | Given that $x-2 b$ is a factor of $a x^{2}-x-a$, find an expression for $a$ in terms of $b$.$\text{ (3 marks)}$ | |

22. | Given that $x+2 b$ is a factor of $a x^{2}+x-2 a$, find an expression for $a$ in term of $b$.$\text{ (3 marks)}$ |

#### ** Answer (2010)**

$\quad\;\,$ | $\,$ | |
---|---|---|

1. | 1,3 | |

2. | $-\frac{3}{4}, 2$ | |

3. | $-\frac{14}{5}, 2$ | |

4. | $6 ; 18$ | |

5. | $-4 ; 12$ | |

6. | $a=2 b-9$ | |

7. | $-7 ;-50$ | |

8. | $-3,1$ | |

9. | $-6,3 \quad$ | |

10. | $-2,-3 \quad$ | |

11. | $\frac{2}{17};\frac{72}{17}$ | |

12. | $-3,3,22 $ | |

13. | $11,-6$ | |

14. | $11,-6$ | |

15. | $-3, \frac{1}{2}, 2 $ | |

16. | $-2, \frac{1}{3}, 3$ | |

17. | $-3,-2,-1,1$ | |

18. | $-5,6,0 ; x(x-2)(x-3)$ | |

19. | $1,3 x-2 $ | |

20. | $2 ; x-5$ | |

21. | $a=\frac{2 b}{4 b^{2}-1}$ | |

22. | $a=\frac{b}{2 b^{2}-1}$ |

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