Mathematics Class IX

Question 1 of 25

1. Question
Which of the following is a factor of $ x^{3}-7x-6 $?
- $ x + 3 $
- $ x + 1 $
- $ x + 4 $
- $ x – 2 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-7x-6 &= x^{3}-x-6x-6 \\ &=x(x^{2}-1)-6(x+1) \\ &= (x)(x-1)(x+1)-6(x+1) \\ &= (x+1)[x(x-1)-6] \\ &= (x+1)(x^{2}-x-6) \end{split}\end{equation*}$

Therefore, [\latexpage] $x+1$ is a factor of the given function.
Question 2 of 25

2. Question
Which of the following is a factor of $ x^{3}-x^{2}-4x+4 $?
- $ x + 4 $
- $ x – 4 $
- $ x – 1 $
- $ x + 1 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-x^{2}-4x+4 &= x^{2}(x-1)-4(x-1) \\ &=(x-1)(x^{2}-4) \\ &= (x-1)(x-2)(x+2) \end{split}\end{equation*}$

Therefore, [\latexpage] $x-1$ is a factor of the given function.
Question 3 of 25

3. Question
Which of the following is a factor of $ x^{3}-2x^{2}-3x $?
- $ x – 1 $
- $ x + 3 $
- $ x + 2 $
- $ x $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-2x^{2}-3x &= x(x^{2}-2x-3) \\ &=x (x^{2}-3x+x-3) \\ &= x(x(x-3)+1(x-3)) \\ &= x(x-3)(x+1)\end{split}\end{equation*}$

Therefore, [\latexpage] $x$ is a factor of the given function.
Question 4 of 25

4. Question
Which of the following is a factor of $ x^{3}-4x $?
- $ x + 2 $
- $ x + 1 $
- $ x + 3 $
- $ x – 1 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-4x &= x(x^{2}-4) \\ &=x (x-2)(x+2) \end{split}\end{equation*}$

Therefore, [\latexpage] $x+2$ is a factor of the given function.
Question 5 of 25

5. Question
Find the value of $ k $ if $ x-2 $ is a factor of $ f(x) = x^{2}-4x+k $.
- 1
- 2
- 3
- 4
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} f(2) &= 0 \\ 2^{2}-4(2)+k &= 0 \\ 4-8+k &= 0 \\ -4+k &= 0 \\ k &= 4 \end{split}\end{equation*}$
Question 6 of 25

6. Question
Find the value of $ p $ if $ x+1 $ is a factor of $ f(x) = px^{2}-8x+3 $.
- 5
- 11
- $ -11 $
- $ -5 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} f(-1) &= 0 \\ p(-1)^{2}-8(-1)+3 &= 0 \\ p+8+3 &= 0 \\ p+11 &= 0 \\ p &= -11 \end{split}\end{equation*}$
Question 7 of 25

7. Question
Find the value of $ g $ if $ x-3 $ is a factor of $ f(x) = x^{2}-2gx+9 $.
- 3
- $ -2 $
- 4
- $ -4 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} f(3) &= 0 \\ 3^{2}-2g(3)+9 &= 0 \\ 9-6g+9 &= 0 \\ 18-6g &= 0 \\ -6g &= -18 \\ g &= 3 \end{split}\end{equation*}$
Question 8 of 25

8. Question
Find the value of $ m $ if $ x+2 $ is a factor of $ f(x) = mx^{2}+5x-10 $.
- 5
- $ -5 $
- 4
- $ -4 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} f(-2) &= 0 \\ m(-2)^{2}+5(-2)-10 &= 0 \\ 4m-10-10 &= 0 \\ 4m-20 &= 0 \\ 4m &= 20 \\ m &= 5 \end{split}\end{equation*}$
Question 9 of 25

9. Question
Find the value of $ m $ if $ x+1 $ is a factor of $ f(x) = 3x^{2}+7x-m $.
- $ -3 $
- $ -4 $
- 3
- 4
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} f(-1) &= 0 \\ 3(-1)^{2}+7(-1)-m &= 0 \\ 3-7-m &= 0 \\ -4-m &= 0 \\ m &= -4 \end{split}\end{equation*}$
Question 10 of 25

10. Question
Factorise the polynomial $ x^{2}-14x+40 $.
- $ (x-10)(x-4) $
- $ (x+10)(x+4) $
- $ (x-10)(x+4) $
- $ (x+10)(x-4) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{2}-14x+40 &= x^{2}-10x-4x+40 \\ &= x(x-10)-4(x-10) \\ &= (x-10)(x-4) \end{split}\end{equation*}$
Question 11 of 25

11. Question
Factorise the polynomial $ x^{2}-13x-30 $.
- $ (x-15)(x-2) $
- $ (x-15)(x+2) $
- $ (x+15)(x+2) $
- $ (x+15)(x-2) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{2}-13x-30 &= x^{2}-15x+2x-30 \\ &= x(x-15)+2(x-15) \\ &= (x-15)(x+2) \end{split}\end{equation*}$
Question 12 of 25

12. Question
Factorise the polynomial $ x^{2}+15x+54 $.
- $ (x+9)(x-6) $
- $ (x-9)(x-6) $
- $ (x+9)(x+6) $
- $ (x-9)(x+6) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{2}+15x+54 &= x^{2}+9x+6x+54 \\ &= x(x+9)+6(x+9) \\ &= (x+9)(x+6) \end{split}\end{equation*}$
Question 13 of 25

13. Question
Factorise the polynomial $ 9x^{2}+6x-8 $.
- $ (3x-4)(3x+2) $
- $ (3x-4)(3x-2) $
- $ (3x+4)(3x+2) $
- $ (3x+4)(3x-2) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} 9x^{2}+6x-8 &= 9x^{2}+12x-6x-8 \\ &= 3x(3x+4)-2(3x+4) \\ &= (3x+4)(3x-2) \end{split}\end{equation*}$
Question 14 of 25

14. Question
Factorise the polynomial $ 2x^{2}+7x-4 $.
- $ (x-4)(2x+s1) $
- $ (x+4)(2x+1) $
- $ (x+4)(2x-1) $
- $ (x-4)(2x-1) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} 2x^{2}+7x-4 &= 2x^{2}+8x-x-4 \\ &= 2x(x+4)-1(x+4) \\ &= (x+4)(2x-1) \end{split}\end{equation*}$
Question 15 of 25

15. Question
Factorise the polynomial $ x^{3}-4x^{2}-x+4 $.
- $ (x-4)(x-1)(x+1) $
- $ (x+4)(x-1)(x+1) $
- $ (x-2)(x-1)(x+2) $
- $ (x-2)(x+1)(x+2) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-4x^{2}-x+4 &= x^{2}(x-4)-1(x-4) \\ &= (x-4)[x^{2}-1] \\ &= (x-4)(x-1)(x+1) \end{split}\end{equation*}$
Question 16 of 25

16. Question
Factorise the polynomial $ x^{3}-2x^{2}-5x+6 $.
- $ (x+1)(x+2)(x-3) $
- $ (x-1)(x+2)(x-3) $
- $ (x-1)(x-2)(x-3) $
- $ (x-1)(x+2)(x+3) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-2x^{2}-5x+6 &= x^{3}-x^{2}-x^{2}+x-6x+6 \\ &= x^{2}(x-1)-x(x-1)-6(x-1) \\ &= (x-1)[x^{2}-x-6] \\ &= (x-1)[x^{2}-3x+2x-6] \\ &= (x-1)[x(x-3)+2(x-3)] \\ &= (x-1)(x+2)(x-3) \end{split}\end{equation*}$
Question 17 of 25

17. Question
Factorise the polynomial $ x^{3}+12x^{2}+44x+48 $.
- $ (x-1)(x-2)(x-3) $
- $ (x-1)(x+2)(x+3) $
- $ (x-1)(x+2)(x-3) $
- $ (x+1)(x+2)(x-3) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}+12x^{2}+44x+48 &= x^{3}+2x^{2}+10x^{2}+20x+24x+48 \\ &= x^{2}(x+2)+10x(x+2)+24(x+2) \\ &= (x+2)[x^{2}+10x+24] \\ &= (x+2)[x^{2}+6x+4x+24] \\ &= (x+2)[x(x+6)+4(x+6)] \\ &= (x+2)(x+6)(x+4) \end{split}\end{equation*}$
Question 18 of 25

18. Question
Factorise the polynomial $ x^{3}-9x^{2}+23x-15 $.
- $ (x+1)(x+3)(x+5) $
- $ (x-1)(x+3)(x-5) $
- $ (x+1)(x-3)(x-5) $
- $ (x-1)(x-3)(x-5) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}-9x^{2}+23x-15 &= x^{3}-x^{2}-8x^{2}+8x+15x-15 \\ &= x^{2}(x-1)-8x(x-1)+15(x-1) \\ &= (x-1)[x^{2}-8x+15] \\ &= (x-1)[x^{2}-3x-5x+15] \\ &= (x-1)[x(x-3)-5(x-3)] \\ &= (x-1)(x-3)(x-5) \end{split}\end{equation*}$
Question 19 of 25

19. Question
Factorise the polynomial $ x^{3}+4x^{2}-9x-36 $.
- $ (x+3)(x-4)(x+4) $
- $ (x-3)(x-4)(x+4) $
- $ (x-4)(x-3)(x+3) $
- $ (x+4)(x-3)(x+3) $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x^{3}+4x^{2}-9x-36 &= x^{2}(x+4)-9(x+4) \\ &= (x+4)[x^{2}-9] \\ &= (x+4)(x-3)(x+3) \end{split}\end{equation*}$
Question 20 of 25

20. Question
Find the product: $ (x+7)(x-8) $.
- $ x^{2}-x-56 $
- $ x^{2}-x+56 $
- $ x^{2}+x-56 $
- $ x^{2}+x+56 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (x+7)(x-8) &= x^{2}-8x+7x-56 \\ &= x^{2}-x-56 \end{split}\end{equation*}$
Question 21 of 25

21. Question
Find the product: $ (x+4)(x+9) $.
- $ x^{2}+13x-36 $
- $ x^{2}+13x+36 $
- $ x^{2}-13x+36 $
- $ x^{2}-13x-36 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (x+4)(x+9) &= x^{2}+9x+4x+36 \\ &= x^{2}+13x+36 \end{split}\end{equation*}$
Question 22 of 25

22. Question
Find the product: $ (3x-5)(x-6) $.
- $ 3x^{2}-23x-30 $
- $ 3x^{2}-23x-30 $
- $ 3x^{2}-23x+30 $
- $ 3x^{2}+23x+30 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (3x-5)(x-6) &= 3x^{2}-18x-5x+30 \\ &= 3x^{2}-23x+30 \end{split}\end{equation*}$
Question 23 of 25

23. Question
Find the product: $ (5x-7)(x+8) $.
- $ 5x^{2}-33x+56 $
- $ 5x^{2}-33x-56 $
- $ 5x^{2}+33x+56 $
- $ 5x^{2}+33x-56 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (5x-7)(x+8) &= 5x^{2}+40x-7x-56 \\ &= 5x^{2}+33x-56 \end{split}\end{equation*}$
Question 24 of 25

24. Question
Expand: $ (x+7)^{3} $
- $ x^{3}+98x^{2}+21x+49 $
- $ x^{3}+14x^{2}+147x+98 $
- $ x^{3}+147x^{2}+21x+343 $
- $ x^{3}+21x^{2}+147x+343 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (x+7)^{3} &= x^{3}+3(x)^{2}(7)+3(x)7^{2}+7^{3} \\ &= x^{3}+21x^{2}+147x+343 \end{split}\end{equation*}$
Question 25 of 25

25. Question
Expand: $ (2m+3n)^{3} $
- $ 4m^{3}+54m^{2}n+36mn^{2}+9n^{3} $
- $ m^{3}+36m^{2}n+54mn^{2}+n^{3} $
- $ 8m^{3}+36m^{2}n+54mn^{2}+27n^{3} $
- $ 8m^{3}+54m^{2}n+36mn^{2}+27n^{3} $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (2m+3n)^{3} &= (2m)^{3}+3(2m)^{2}(3n)+3(2m)(3n)^{2}+(3n)^{3} \\ &= 8m^{3}+36m^{2}n+54mn^{2}+27n^{3} \end{split}\end{equation*}$