# Remainder Theorem (Myanmar Exam Board)

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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)

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

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)

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

$\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)