Grade 11 Mathematics Myanmar
2. Using the remainder theorem, find the remainder when
3. Find the remainder when
4. When $x^{3}+3 x^{2}-m x+4$ is divided by $x-2$, the remainder is $m+3$. Find the value of $m$.
5. The polynomial $x^{3}+a x^{2}+b x-3$ leaves a remainder of 27 when divided by $x-2$ and a remainder of 3 when divided by $x+1$. Calculate the remainder when the polynomial is divided by $x-1$.
6. The expressions $x^{3}-7 x+6$ and $x^{3}-x^{2}-4 x+24$ have the same remainder when divided by $x+p$. Find the possible values of $p$.
7. 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 ' $b$ '.
8. 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 $x-2$.
9. When the expression $x^{3}+a x^{2}+4$ is divided by $x+1$ the remainder is 6 greater than the remainder when it is divided by $x-2$. Find the value of $a$.
10. What number should be subtracted from $3 x^{3}-5 x^{2}+6 x$ so that on dividing it by $x-3$, the remainder is 8 ?
11. 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.
12. When the polynomials $a x^{3}+5 x^{2}+3 x+4$ and $3 x^{3}+9 x^{2}+a x-6$ are divided by $x+3$, the remainders are $A$ and $B$ respectively. Find the values of $a$ if $A+B=4$.
13. The remainder when $a x^{3}+b x^{2}+2 x+3$ is divided by $x-1$ is twice that when it is divided by $x+1$, show that $b=3 a+3$.
14. 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 $k$.
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 $x+a$. Calculate the possible values of $a$.
16. 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 $x-2$.
Answers Exercise 1.1
1(a) $Q(x)=3x+1, R=2$
(b) $Q(x)=-2x-2,R=5$
(c) $Q(x)=4x^2+12x+38 ,R=111$
(d) $Q(x)=2x^2+2,R=0$
(e) $Q(x)=x^3+\sqrt 3x^2-3x-3\sqrt 3,R=0$
2(a) $R=7$
(b) $R=-10$
(c) $R=6$
(d) $R=-7$
(e) $R=6$
3(a) $R=5$
(b) $R=\frac{-27}{4}$
4 $m=7$
5 $a=6,b=-1,R=3$
6 $p=-6$ or $p=3$
7 $a=b+3$
8 $a=\frac{2}{17},R=\frac{72}{17}$
9 $a=-5$
10 $a=46$
11 $a=4,b=-12,c=9,f(x)=(2x-3)^2$
12 $a=1$
13 $b=3a+3$
14 $k=-5$
15 $a=-\frac{7}{9}$ or $a=-1$
16 $a=3,b-2,R=23$
Exercise $1.1$
1. Using synthetic division, find the remainder and quotient when| (a) $3 x^{2}-2 x+1$ is divided by $x-1$ (b) $3-4 x-2 x^{2}$ is divided by $x+1$ (c) $4 x^{3}+2 x-3$ is divided by $x-3$ (d) $2 x^{3}+x^{2}+2 x+1$ is divided by $x+\frac{1}{2}$ (e) $x^{4}-6 x^{2}+9$ is divided by $x-\sqrt{3}$ |
2. Using the remainder theorem, find the remainder when
| (a) $5 x^{2}+7 x+9$ is divided by $x+1$. (b) $-x^{3}+3 x^{2}-7 x$ is divided by $x-2$. (c) $2 x^{3}+3 x^{2}+5$ is divided by $x-\frac{1}{2}$. (d) $x^{6}-6 x^{4}+12 x^{2}-15$ is divided by $x+\sqrt{2}$. (e) $x^{4}-2 x^{2}+6$ is divided by $x$. |
3. Find the remainder when
| (a) $6 x^{2}+x-7$ is divided by $2 x+3$. (b) $10 x^{4}+5 x^{3}+4 x^{2}-9$ is divided by $2 x-1$. |
4. When $x^{3}+3 x^{2}-m x+4$ is divided by $x-2$, the remainder is $m+3$. Find the value of $m$.
5. The polynomial $x^{3}+a x^{2}+b x-3$ leaves a remainder of 27 when divided by $x-2$ and a remainder of 3 when divided by $x+1$. Calculate the remainder when the polynomial is divided by $x-1$.
6. The expressions $x^{3}-7 x+6$ and $x^{3}-x^{2}-4 x+24$ have the same remainder when divided by $x+p$. Find the possible values of $p$.
7. 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 ' $b$ '.
8. 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 $x-2$.
9. When the expression $x^{3}+a x^{2}+4$ is divided by $x+1$ the remainder is 6 greater than the remainder when it is divided by $x-2$. Find the value of $a$.
10. What number should be subtracted from $3 x^{3}-5 x^{2}+6 x$ so that on dividing it by $x-3$, the remainder is 8 ?
11. 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.
12. When the polynomials $a x^{3}+5 x^{2}+3 x+4$ and $3 x^{3}+9 x^{2}+a x-6$ are divided by $x+3$, the remainders are $A$ and $B$ respectively. Find the values of $a$ if $A+B=4$.
13. The remainder when $a x^{3}+b x^{2}+2 x+3$ is divided by $x-1$ is twice that when it is divided by $x+1$, show that $b=3 a+3$.
14. 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 $k$.
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 $x+a$. Calculate the possible values of $a$.
16. 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 $x-2$.
Answers Exercise 1.1
1(a) $Q(x)=3x+1, R=2$
(b) $Q(x)=-2x-2,R=5$
(c) $Q(x)=4x^2+12x+38 ,R=111$
(d) $Q(x)=2x^2+2,R=0$
(e) $Q(x)=x^3+\sqrt 3x^2-3x-3\sqrt 3,R=0$
2(a) $R=7$
(b) $R=-10$
(c) $R=6$
(d) $R=-7$
(e) $R=6$
3(a) $R=5$
(b) $R=\frac{-27}{4}$
4 $m=7$
5 $a=6,b=-1,R=3$
6 $p=-6$ or $p=3$
7 $a=b+3$
8 $a=\frac{2}{17},R=\frac{72}{17}$
9 $a=-5$
10 $a=46$
11 $a=4,b=-12,c=9,f(x)=(2x-3)^2$
12 $a=1$
13 $b=3a+3$
14 $k=-5$
15 $a=-\frac{7}{9}$ or $a=-1$
16 $a=3,b-2,R=23$
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