Algebraic manipulation

1. Simplify 9x - 4 + 3x + 11.

Answer C: 12x + 7

The like x-terms give 9x + 3x = 12x, while the constants give -4 + 11 = 7. The simplified expression is 12x + 7.

2. Simplify 5a^2 - 3ab + 7 + 2a^2 + 8ab - 10.

Answer B: 7a^2 + 5ab - 3

The coefficients combine as 5 + 2 = 7 for a^2, -3 + 8 = 5 for ab, and 7 - 10 = -3. Therefore the result is 7a^2 + 5ab - 3.

3. Simplify 6xy - 2xy + 3yx + 5x^2.

Answer D: 7xy + 5x^2

Because xy = yx, the first three coefficients combine to 6 - 2 + 3 = 7. The term 5x^2 is unlike xy, so the answer is 7xy + 5x^2.

4. Expand 4x(3x - 5y).

Answer A: 12x^2 - 20xy

The two products are 4x times 3x = 12x^2 and 4x times -5y = -20xy. Hence the expanded expression is 12x^2 - 20xy.

5. Expand -3a(2a^2 - 5a + 4).

Answer C: -6a^3 + 15a^2 - 12a

Multiplying term by term gives -6a^3, +15a^2 and -12a. The double negative creates the positive middle term, so the result is -6a^3 + 15a^2 - 12a.

6. Expand (x + 5)(x - 2).

Answer B: x^2 + 3x - 10

The products are x^2, -2x, 5x and -10. Combining -2x + 5x gives 3x, so the expansion is x^2 + 3x - 10.

7. Expand (2x - 3)(x + 4).

Answer D: 2x^2 + 5x - 12

The four products are 2x^2, 8x, -3x and -12. Since 8x - 3x = 5x, the fully expanded result is 2x^2 + 5x - 12.

8. Expand (2x + y)(3x - 4y).

Answer A: 6x^2 - 5xy - 4y^2

The products are 6x^2, -8xy, 3xy and -4y^2. The cross-products combine to -5xy, giving 6x^2 - 5xy - 4y^2.

9. Expand (x - 1)(x + 2)(x + 3).

Answer C: x^3 + 4x^2 + x - 6

First, (x - 1)(x + 2) = x^2 + x - 2. Multiplying by x + 3 gives x^3 + 4x^2 + x - 6 after collecting like terms.

10. Factorise fully 14x^2y - 21xy^2.

Answer B: 7xy(2x - 3y)

Both terms share 7xy. Dividing them by 7xy leaves 2x and -3y, so the full factorisation is 7xy(2x - 3y).

11. Factorise fully 12a^3 + 18a^2 + 6a.

Answer D: 6a(2a + 1)(a + 1)

Extracting 6a gives 6a(2a^2 + 3a + 1). The quadratic is (2a + 1)(a + 1), so the fully factorised answer is 6a(2a + 1)(a + 1).

12. Factorise px + qx + 3py + 3qy by grouping.

Answer A: (p + q)(x + 3y)

The expression becomes x(p + q) + 3y(p + q). Extracting the common binomial p + q gives the factorisation (p + q)(x + 3y).

13. Factorise fully 25x^2 - 49y^2.

Answer C: (5x - 7y)(5x + 7y)

This is (5x)^2 - (7y)^2. Applying A^2 - B^2 = (A - B)(A + B) gives (5x - 7y)(5x + 7y).

14. Factorise 9a^2 + 24ab + 16b^2.

Answer B: (3a + 4b)^2

The end terms are (3a)^2 and (4b)^2, and 2(3a)(4b) = 24ab. Therefore the trinomial is the perfect square (3a + 4b)^2.

15. Factorise x^2 + 9x + 20.

Answer D: (x + 4)(x + 5)

The integers 4 and 5 multiply to 20 and add to 9. Thus x^2 + 9x + 20 factorises as (x + 4)(x + 5).

16. Factorise x^2 - 3x - 28.

Answer A: (x - 7)(x + 4)

The pair -7 and 4 has product -28 and sum -3. Therefore the factorisation is (x - 7)(x + 4), which expands to the original quadratic.

17. Factorise fully 6x^2 + 11x + 3.

Answer C: (3x + 1)(2x + 3)

Using 9 and 2 gives 6x^2 + 9x + 2x + 3. Grouping produces 3x(2x + 3) + 1(2x + 3) = (3x + 1)(2x + 3).

18. Factorise fully 8x^2 - 2x - 3.

Answer B: (4x - 3)(2x + 1)

Rewrite the middle term to get 8x^2 + 4x - 6x - 3. Grouping gives 4x(2x + 1) - 3(2x + 1), hence (4x - 3)(2x + 1).

19. Factorise fully 2x^3 - 5x^2 - 3x.

Answer D: x(2x + 1)(x - 3)

After extracting x, the quadratic factorises as (2x + 1)(x - 3), since its cross-terms are -6x + x = -5x. The full answer is x(2x + 1)(x - 3).

20. Factorise fully 3x^3 - 12x.

Answer A: 3x(x - 2)(x + 2)

Extracting 3x gives 3x(x^2 - 4). Since x^2 - 4 = (x - 2)(x + 2), the fully factorised expression is 3x(x - 2)(x + 2).

21. Write x^2 + 8x + 3 in completed-square form.

Answer C: (x + 4)^2 - 13

Because (x + 4)^2 = x^2 + 8x + 16, subtract 13 to make the constant 3. Thus x^2 + 8x + 3 = (x + 4)^2 - 13.

22. Write x^2 - 6x + 11 in completed-square form.

Answer B: (x - 3)^2 + 2

The square (x - 3)^2 expands to x^2 - 6x + 9. Adding 2 gives the required constant 11, so the completed-square form is (x - 3)^2 + 2.

23. Write 2x^2 + 12x - 5 in completed-square form.

Answer D: 2(x + 3)^2 - 23

Write 2(x^2 + 6x) - 5 = 2[(x + 3)^2 - 9] - 5. Simplifying gives 2(x + 3)^2 - 23.

24. Write -3x^2 + 12x + 1 in completed-square form.

Answer A: -3(x - 2)^2 + 13

Write -3(x^2 - 4x) + 1 = -3[(x - 2)^2 - 4] + 1. This simplifies to -3(x - 2)^2 + 13.