Multiple Choice

1. Factor the expression $x^2 - 9$.

   $(x^2 - 3)(x^2 + 3)$

   $(x - 9)(x + 9)$

   $(x - 3)^2$

   $(x - 3)(x + 3)$

2. Factor the expression $4y^2 - 25$.

   $(4y - 5)(4y + 5)$

   $(2y - 25)(2y + 25)$

   $(2y - 5)(2y + 5)$

   $(4y - 5)(y + 5)$

3. Completely factor the expression $3x^2 - 75$.

   $3(x - 25)(x + 25)$

   $3(x - 5)(x + 5)$

   $(3x - 15)(x + 5)$

   $3(x^2 - 25)$

4. Factor the expression $16m^2 - 49n^2$.

   $(16m - 49n)(m + n)$

   $(4m - 7n)(4m + 7n)$

   $(16m - 7n)(m + 7n)$

   $(8m - 7n)(8m + 7n)$

5. Factor the expression $1 - 100p^2$.

   $(1 - 10p)(1 + 10p)$

   $(-1 + 10p)(-1 - 10p)$

   $(10p - 1)(10p + 1)$

   $(1 - 50p)(1 + 50p)$

6. Factor the expression $x^4 - 1$ completely.

   $(x - 1)^2(x + 1)^2$

   $(x - 1)(x + 1)(x^2 + 1)$

   $(x^2 - 1)^2$

   $(x^2 - 1)(x^2 + 1)$

7. Which of the following expressions CANNOT be factored using the difference of two squares method?

   $81 - y^2$

   $m^2 - 121$

   $y^2 + 4$

   $4x^2 - 16$

8. Completely factor the expression $18x^3 - 50xy^2$.

   $2(3x^2 - 5y)(3x^2 + 5y)$

   $2x(3x - 5y)(3x + 5y)$

   $2x(3x - 5y)(3x - 5y)$

   $2x(9x^2 - 25y^2)$

9. Factor the expression $a^{10} - b^{12}$.

   $(a^{10} - b^{12})(a^{10} + b^{12})$

   $(a^5 - b^6)^2$

   $(a^2 - b^2)(a^5 + b^6)$

   $(a^5 - b^6)(a^5 + b^6)$

10. The expression $81z^2 - 64$ is factored into $(9z - 8)(9z + k)$. What is the value of $k$?

    $64$

    $-8$

    $9$

    $8$

11. Factorise the expression $5 - x$ using the difference of two squares method involving radicals.

    $(\sqrt{5} - x)(\sqrt{5} + x)$

    $(5 - \sqrt{x})(5 + \sqrt{x})$

    $(5 - x)(5 + x)$

    $(\sqrt{5} - \sqrt{x})(\sqrt{5} + \sqrt{x})$

12. Determine the complete factorisation of $9a - 16b$ using the difference of two squares method.

    $(9\sqrt{a} - 16\sqrt{b})(9\sqrt{a} + 16\sqrt{b})$

    $(3\sqrt{a} - 4\sqrt{b})(3\sqrt{a} + 4\sqrt{b})$

    $(\sqrt{9a} - \sqrt{16b})(\sqrt{9a} + \sqrt{16b})$

    $(3a - 4b)(3a + 4b)$

13. Factorise the expression $4 - 3y$.

    $(2 - 3y)(2 + 3y)$

    $(2 - \sqrt{3y})(2 + \sqrt{3y})$

    $(\sqrt{4y} - \sqrt{3})(\sqrt{4y} + \sqrt{3})$

    $(2 - \sqrt{3}y)(2 + \sqrt{3}y)$

14. If the difference of two squares factorisation of $50 - 2x$ is $k(5 - \sqrt{x})(5 + \sqrt{x})$, find the value of $k$.

    $k=1$

    $k=5$

    $k=2$

    $k=50$

15. Factorise $11 - 2z$ completely using the difference of two squares.

    $(\sqrt{11} - \sqrt{z})(\sqrt{11} + \sqrt{z})$

    $(11 - \sqrt{2z})(11 + \sqrt{2z})$

    $(\sqrt{11} - 2\sqrt{z})(\sqrt{11} + 2\sqrt{z})$

    $(\sqrt{11} - \sqrt{2z})(\sqrt{11} + \sqrt{2z})$

16. Which of the following is the correct factorisation of $x^2 - 5$?

    $(\sqrt{x^2} - \sqrt{5})(\sqrt{x^2} + \sqrt{5})$

    $(x - \sqrt{5})(x + \sqrt{5})$

    $(x - 5)(x + 5)$

    $(\sqrt{x} - 5)(\sqrt{x} + 5)$

17. Simplify the expression that results from multiplying the factors $(2\sqrt{3} - \sqrt{y})(2\sqrt{3} + \sqrt{y})$.

    $6 - y$

    $4\sqrt{3} - y$

    $12 + y$

    $12 - y$

18. The expression $6 - 15w$ can be written in the form $3(2 - 5w)$. If we apply DOTS to the bracketed term, what is the complete factorisation?

    $3(2 - \sqrt{5w})(2 + \sqrt{5w})$

    $3(\sqrt{2} - \sqrt{5w})(\sqrt{2} + \sqrt{5w})$

    $3(4 - 25w^2)$

    $(3\sqrt{2} - 3\sqrt{5w})(3\sqrt{2} + 3\sqrt{5w})$

19. What is the original expression if its difference of two squares factorisation is $(3a - \sqrt{10})(3a + \sqrt{10})$?

    $3a^2 - 10$

    $9a^2 - 10$

    $9a^2 - 5$

    $9a - 10$

20. Factorise $1 - 20x$ using the difference of two squares.

    $(1 - \sqrt{20x})(1 + \sqrt{20x})$

    $(\sqrt{1} - 2\sqrt{5x})(\sqrt{1} + 2\sqrt{5x})$

    $(1 - 20\sqrt{x})(1 + 20\sqrt{x})$

    $(1 - 4\sqrt{5x})(1 + 4\sqrt{5x})$

21. Factorise completely the expression $2x^2 - 50$.

    $2(x^2-25)$

    $2(x-5)(x+5)$

    $(2x-10)(x+5)$

    $(x-5)(2x+10)$

22. Factorise completely: $(x+3)^2 - 9$.

    $x(x+6)$

    $(x^2+6x+18)$

    $(x+6)(x-6)$

    $(x+9)(x-9)$

23. Factorise completely: $(2x-1)^2 - (x+4)^2$.

    $3(x-5)(x+1)$

    $(x-5)(3x-3)$

    $3(x-5)(x+3)$

    $(x+3)(3x-5)$

24. When applying the difference of squares rule, $A^2 - B^2 = (A - B)(A + B)$, to the given expression, calculate the resulting $(A + B)$ factor.

    $x - 5$

    $x + 5$

    $3x + 3$

    $3x - 3$

25. Expand and simplify the expression $(x+5)^2 - 16$ into the standard quadratic form $ax^2 + bx + c$.

    $x^2 + 25x + 9$

    $x^2 + 10x + 9$

    $x^2 + 5x - 16$

    $x^2 + 10x + 41$

26. The expression $4(x+3)^2 - 100$ is equivalent to $4(x-A)(x+B)$. What is the value of $A+B$?

    $10$

    $4$

    $-6$

    $-4$

27. Factor the quadratic expression $P(x) = (x-6)^2 - 49$ completely.

    $(x-13)(x-1)$

    $(x-13)(x+1)$

    $(x+13)(x-1)$

    $(x+6)(x-7)$

28. Factorise the expression $(x+3)^2 - 49$.

    $(x+10)(x+4)$

    $(x-7)(x+7)$

    $(x+10)(x-4)$

29. Fully factorise the expression $P = (x-1)^2 - 9$.

    $(x-4)(x+2)$

    $(x-10)(x+8)$

    $(x-4)(x-2)$

    $(x^2 - 10)$

30. What is the factorised form of $(2x+1)^2 - 25$?

    $(2x^2 - 24)$

    $(4x-24)(2x+26)$

    $(2x+6)(2x-4)$

    $(2x+6)(2x+4)$

31. Factorise the expression $100 - (y-2)^2$.

    $(10 + y)(10 - y)$

    $(12 - y)(8 + y)$

    $(102 - y)(98 + y)$

    $(8 + y)(12 - y)$

32. Fully factorise the expression $2(x+1)^2 - 50$.

    $2(x^2 - 24)$

    $2(x+6)(x-4)$

    $(2x+12)(x-4)$

    $2(x+4)(x-6)$

33. Factorise the algebraic expression $(x+y)^2 - z^2$.

    $(x+y-z)(x-y+z)$

    $(x+y+z)(x-y-z)$

    $(x^2+y^2-z^2)$

    $(x+y+z)(x+y-z)$

34. Factorise the expression $(3x+4)^2 - \frac{1}{4}$.

    $(3x + 4.5)(3x + 3)$

    $(3x + \frac{9}{2})(3x + \frac{7}{2})$

    $(3x - \frac{1}{2})(3x + \frac{1}{2})$

    $(3x + 4)(\frac{1}{2})$