Question 1
LEVEL 6
Expand and simplify (a + 1)(a + 2)(a - 3), then choose the correct answer from the list below.
A: a^3 + 6a^2 - 7a - 6
B: a^3 - 7a - 6
C: a^3 + 6a^2 + 6a - 7
D: a^3 - 7a + 6
CORRECT ANSWER: B: a^3 - 7a - 6
WORKED SOLUTION:
Firstly, we expand the first two brackets as a normal double bracket expansion.
= a^2 + 2a + a + 2
= a^2 + 3a + 2
Next, we multiply the result of this by the third bracket. See:
= a^3 + 3a^2 + 2a - 3a^2 - 9a - 6
= a^3 - 7a - 6.
Question 2
LEVEL 6
Which one of the following answers is the correct expansion of (x - 3)^3?
A: x^3 - 9x^2 + 27x - 27
B: x^3 - 27
C: x^3 + 9x^2 - 27
D: x^3 - 9x^2 + 27x + 27
CORRECT ANSWER: A: x^3 - 9x^2 + 27x - 27
WORKED SOLUTION:
Firstly, we recognise that
(x - 3)^3 = (x - 3)(x - 3)(x - 3)
Then, we can begin the expansion by considering just the first two brackets and expanding them as a normal double bracket.
= x^2 - 3x - 3x + 9
= x^2 - 6x + 9
Then, we can multiply the result of this expansion by the third bracket. See:
= x^3 - 6x^2 + 9x - 3x^2 + 18x - 27
= x^3 - 9x^2 + 27x - 27.
Question 3
LEVEL 6
From the list below, choose the correct expansion of (p + 5q)(q - p)(q+p).
A: 5q^3 - 4pq^2 - p^3
B: 5q^3 - 4p^2q - p^3
C: 5q^3 - 5p^2q + pq^2 - p^3
D: 5q^3 - 5p^2q + pq^2 + p^3
CORRECT ANSWER: C: 5q^3 - 5p^2q + pq^2 - p^3
WORKED SOLUTION:
= (p \times q) + (p \times -p) + (5q \times q) + (5q \times -p)
= pq - p^2 + 5q^2 - 5pq
= 5q^2 - 4pq - p^2.
(p+5q)(q-p)(q+p) = (5q^2 - 4pq - p^2)(q+p)
= 5q^3 - 5p^2q + pq^2 - p^3Question 4
LEVEL 6
Which one of the following options is the correct result of expanding and simplifying (2m - 3)(2m + 3)(m - 7)?
A: 4m^3 + 28m^2 + 9m + 63
B: 4m^3 - 28m^2 + 9m + 63
C: 4m^3 - 28m^2 - 9m + 63
D: 4m^3 - 28m^2 - 9m - 63
CORRECT ANSWER: C: 4m^3 - 28m^2 - 9m + 63
WORKED SOLUTION:
Firstly, we expand the first two brackets on their own as a normal double bracket expansion. We can observe that it is a case of the “difference of two squares”, so we get
(2m - 3)(2m + 3) = 4m^2 - 9.
Then, we multiply this result by the third bracket. See:
= 4m^3 - 28m^2 - 9m + 63.
Question 5
LEVEL 6
Expand and simplify (p + 5q)(q - p)(q + 8), and then choose the correct answer from the options below.
A: 5q^3 - pq^2 - 4p^{2}q + 40q^2 - 8p^2 - 32pq
B: 5q^3 + pq^2 + 4p^{2}q - 40q^2 + 8p^2 + 32pq
C: 5q^3 - p^{2}q - 4pq^2 + 40q^2 - 8p^2 - 32pq
D: 5q^3 - p^{2}q - 4pq^2 - 40q^2 - 8p^2 + 32pq
CORRECT ANSWER: C: 5q^3 - p^{2}q - 4pq^2 + 40q^2 - 8p^2 - 32pq
WORKED SOLUTION:
Firstly, we must expand the first two brackets as a normal double bracket expansion. We get:
= pq - p^2 + 5q^2 - 5pq
= 5q^2 - p^2 - 4pq.
Then, we can multiply the result of this expansion by the third bracket, (q + 8). See:
= 5q^3 - p^{2}q - 4pq^2 + 40q^2 - 8p^2 - 32pq.