Polynomials – Expansion & Factorisation Quiz - ICSE Class 10 Mathematics

Free practice quiz: Polynomials – Expansion & Factorisation (ICSE Class 10 Mathematics). 42 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 10 Mathematics practice quiz has 42 multiple-choice questions on Polynomials – Expansion & Factorisation, and 42 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. If $x + \frac{1}{x} = 3$, find the value of $x^3 + \frac{1}{x^3}$.

   1. 27
   2. 24
   3. 21
   4. 18
   Show answer

   Answer: B. 24

   We know that $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$. Let $a=x$ and $b=\frac{1}{x}$. Then $(x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3x(\frac{1}{x})(x + \frac{1}{x})$. Substituting the given value $x + \frac{1}{x} = 3$, we get $3^3 = x^3 + \frac{1}{x^3} + 3(1)(3)$. So, $27 = x^3 + \frac{1}{x^3} + 9$. Therefore, $x^3 + \frac{1}{x^3} = 27 - 9 = 24$.

2. Factorize the expression $x^3 - 6x^2 + 11x - 6$.

   1. $(x-1)(x-2)(x-3)$
   2. $(x+1)(x-2)(x-3)$
   3. $(x-1)(x+2)(x-3)$
   4. $(x-1)(x-2)(x+3)$
   Show answer

   Answer: A. $(x-1)(x-2)(x-3)$

   Let $P(x) = x^3 - 6x^2 + 11x - 6$. We can use the Rational Root Theorem to find possible rational roots. Possible rational roots are divisors of the constant term -6, which are $\pm1, \pm2, \pm3, \pm6$. Let's test $x=1$: $P(1) = 1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0$. So, $(x-1)$ is a factor. Let's test $x=2$: $P(2) = 2^3 - 6(2)^2 + 11(2) - 6 = 8 - 6(4) + 22 - 6 = 8 - 24 + 22 - 6 = 0$. So, $(x-2)$ is a factor. Let's test $x=3$: $P(3) = 3^3 - 6(3)^2 + 11(3) - 6 = 27 - 6(9) + 33 - 6 = 27 - 54 + 33 - 6 = 0$. So, $(x-3)$ is a factor. Since we have found three roots for a cubic polynomial, the factorization is $(x-1)(x-2)(x-3)$.

3. Factorize $x^4 - 16y^4$.

   1. $(x^2-4y^2)(x^2+4y^2)$
   2. $(x-2y)(x+2y)(x^2+4y^2)$
   3. $(x^2-2y^2)(x^2+2y^2)$
   4. $(x-2y)(x+2y)(x^2-4y^2)$
   Show answer

   Answer: B. $(x-2y)(x+2y)(x^2+4y^2)$

   The expression $x^4 - 16y^4$ can be written as $(x^2)^2 - (4y^2)^2$. This is a difference of squares, so it factors as $(x^2 - 4y^2)(x^2 + 4y^2)$. The term $x^2 - 4y^2$ can be further factored as $(x)^2 - (2y)^2$, which is also a difference of squares. So, $x^2 - 4y^2 = (x - 2y)(x + 2y)$. Therefore, the complete factorization is $(x - 2y)(x + 2y)(x^2 + 4y^2)$.

4. If $a+b=7$ and $a^2+b^2=25$, find the value of $ab$.

   1. 10
   2. 12
   3. 14
   4. 16
   Show answer

   Answer: B. 12

   We use the identity $(a+b)^2 = a^2+b^2+2ab$. We are given $a+b=7$ and $a^2+b^2=25$. Substitute these values into the identity: $7^2 = 25 + 2ab$. $49 = 25 + 2ab$. $49 - 25 = 2ab$. $24 = 2ab$. $ab = \frac{24}{2} = 12$.

5. Factorize $x^3+1$.

   1. $(x+1)(x^2-x+1)$
   2. $(x-1)(x^2+x+1)$
   3. $(x+1)(x^2+x+1)$
   4. $(x-1)(x^2-x+1)$
   Show answer

   Answer: A. $(x+1)(x^2-x+1)$

   This is a sum of cubes expression, $a^3 + b^3 = (a+b)(a^2-ab+b^2)$. Here, $a^3 = x^3 \implies a=x$, and $b^3 = 1^3 \implies b=1$. Substituting into the formula: $(x+1)((x)^2 - (x)(1) + (1)^2)$. $= (x+1)(x^2 - x + 1)$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. If $a+b+c=0$, then find the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$.
2. Factorize: $x^2 + 4xy + 4y^2 - 9z^2$.
3. Expand $(a+b+c)^2$.
4. If $x=\frac{1}{2-\sqrt{3}}$, find the value of $x^2-4x+1$.
5. Factorize $12x^2 - 7x - 10$.
6. What is the coefficient of $x^3$ in the expansion of $(2x-1)(x^2+3x-2)$?
7. If $a-b=3$ and $a^3-b^3=117$, find the value of $a^2+b^2$.
8. Factorize: $x^4 + x^2y^2 + y^4$.
9. If $a+b=10$ and $a^2+b^2=52$, find the value of $ab$.
10. Expand $(x+2y)^3$.
11. Factorize $a^2 - b^2 - 2bc - c^2$.
12. If $x = 3+\sqrt{8}$, find $x^2 - 6x + 1$.
13. Find the value of $(x+y)^2 - (x-y)^2$.
14. If $a-b=2$ and $a^2+b^2=20$, find the value of $a^3-b^3$.
15. If $x + \frac{1}{x} = 4$, find the value of $x^2 + \frac{1}{x^2}$.
16. Expand $(a-b-c)^2$.
17. Factorize $x^3 - 12x^2 + 44x - 48$.
18. Factorize $x^3 + 3x^2 + 3x + 1$.
19. If $a+b+c=9$ and $a^2+b^2+c^2=27$, find the value of $ab+bc+ca$.
20. Factorize $8x^3 - 27y^3$.
21. If $x - \frac{1}{x} = 3$, find the value of $x^3 - \frac{1}{x^3}$.
22. Factorize $x^3 - 3x^2 + 3x - 1$.
23. Factorize $x^3 - 2x^2 - x + 2$.
24. If $a+b+c=0$, find the value of $\frac{a^3+b^3+c^3}{abc}$.
25. Expand $(2a-3b)^3$.
26. Factorize $x^3 - y^3 - 3xy(x-y)$.
27. If $a+b+c=6$, $ab+bc+ca=11$, $abc=6$, find the value of $a^3+b^3+c^3$.
28. Factorize $x^4 + x^2 + 1$.
29. If $x = \frac{\sqrt{5}+1}{2}$, find the value of $x^2 - x - 1$.
30. If $a+b=3$ and $a^3+b^3=27$, find the value of $a^2+b^2$.
31. Factorize $a^3+b^3+c^3-3abc$.
32. If $x^2 - 4x + 1 = 0$, find the value of $x + \frac{1}{x}$.
33. Expand $(x-2y+3z)^2$.
34. Factorize $27x^3 + 1$.
35. Factorize $125x^3 + 8y^3$.
36. If $x+\frac{1}{x}=5$, find the value of $x^3+\frac{1}{x^3}$.
37. Factorize $a^3 - b^3 + 1 - 3ab$.

More quizzes

• Statistics and Probability▶ Play
• Trigonometry – Ratios, Identities, Heights & Distances▶ Play
• Commercial Mathematics & Profit‑Loss▶ Play
• Compound Interest▶ Play
• Coordinate Geometry – Straight Line▶ Play
• Indices and Logarithms▶ Play

Browse all Mathematics quizzes

More to play