Expand 5x(x2 + 3)
Answer
Expanding and factorising — Notes & Worked Examples Show / Hide
Explanation:
To expand a single bracket, multiply the term outside the bracket by every term inside the bracket. When expanding and simplifying two separate brackets, be careful with a negative sign in front of the second bracket (e.g. −2(x − 3) = −2x + 6).
Factorising is the reverse of expanding: find the highest common factor (HCF) of the numbers and the letters, write it outside a bracket, and divide each term by that HCF to find what goes inside. Check by expanding back out.
Example 1: Expand and simplify 4(2x + 3) − 3(x − 5)
Solution:
Expand 1st bracket: 4(2x + 3) = 8x + 12
Expand 2nd bracket: −3(x − 5) = −3x + 15
Collect like terms: 8x − 3x + 12 + 15
Answer: 5x + 27
Example 2: Factorise fully 12x2y − 18xy3
Solution:
HCF of 12 and 18 is 6.
Common letters: xy → full HCF = 6xy
Divide each term by 6xy: 2x and −3y2
Answer: 6xy(2x − 3y2)
Factorise fully 50x + 100
Answer
Worked Solution & Answer
The highest common factor of 50 and 100 is 50: M1
50x + 100 = 50(x + 2) A1
50x + 100 = 50(x + 2) A1
Answer: Answer 50(x + 2)
Multiply out 5x(3x - 2)
Answer
Worked Solution & Answer
5x × 3x - 5x × 2 = 15x2 - 10x M1 A1
Answer: Answer 15x2 - 10x
(a) Expand 5(2m – 3) (1) (b) Factorise 3n + 12 (1)
(a) Expand 5(2m - 3) [1 mark]
(b) Factorise 3n + 12 [1 mark]
Worked Solution & Answer
(a)
5 × 2m – 5 × 3 = 10m – 15 B1
Answer (a): 10m - 15
(b)
3(n + 4)
Answer (b): 3(n + 4)
Work out
(a) Factorise 6x - 15 (1)
(b) Factorise m2 + 5m (1)
(a) Factorise 6x - 15 (1)
(b) Factorise m2 + 5m (1)
(a) Factorise 6x - 15 [1 mark]
(b) Factorise m2 + 5m [1 mark]
Worked Solution & Answer
(a)
3(2x – 5)
Answer (a): 3(2x - 5)
(b)
m(m + 5)
Answer (b): m(m + 5)
(a) Factorise 4m + 12. (1) (b) Complete: 5y is a [term] in the [expression] 3x + 5y. (2)
(a) Factorise 4m + 12 [1 mark]
(b) Choose two words from the box above to make this statement correct. [2 marks]
Worked Solution & Answer
(a)
4(m + 3)
Answer (a): 4(m + 3)
(b)
5y is a term in the expression 3x + 5y M1 A1
Answer (b): term, expression
(a) Factorise 5 – 10m (1) (b) Factorise fully 2a²b + 6ab² (2)
(a) Factorise 5 - 10m [1 mark]
(b) Factorise fully 2a2b + 6ab2 [2 marks]
Worked Solution & Answer
(a)
Common factor 5 → 5(1 – 2m) B1
Answer (a): 5(1 - 2m)
(b)
HCF is 2ab → 2ab(a + 3b) M1 A1
Answer (b): 2ab(a + 3b)
(a) Factorise fully 15w² – 5w (2)
(b) On the number line, show –2 < x ⩽ 4 (2)
(b) On the number line, show –2 < x ⩽ 4 (2)
(a) Factorise fully 15w2 - 5w [2 marks]
(b) On the number line below, show the set of values of x for which –2 < x ≤ 4 [2 marks]
Worked Solution & Answer
(a)
Difference of two squares: a2 - b2 = (a + b)(a - b) M1
e.g. m2 - 9 = (m + 3)(m - 3)
Answer: 5w(3w - 1) A1
e.g. m2 - 9 = (m + 3)(m - 3)
Answer: 5w(3w - 1) A1
Answer (a): 5w(3w - 1)
(b)
Open circle at –2, closed circle at 4, joined by a line M1 A1
Answer (b): Open circle at –2, closed circle at 4, joined by a line
Work out
(a) Simplify 2e - 3f + 4e - 7f (2) (b) Expand and simplify 5(4x + 3) - (3x - 1) (2) (c) Factorise fully 4p + 6pq (2)
(a) Simplify 2e - 3f + 4e - 7f (2) (b) Expand and simplify 5(4x + 3) - (3x - 1) (2) (c) Factorise fully 4p + 6pq (2)
(a) Simplify 2e - 3f + 4e - 7f [2 marks]
(b) Expand and simplify 5(4x + 3) - (3x - 1) [2 marks]
(c) Factorise fully 4p + 6pq [2 marks]
Worked Solution & Answer
(a)
(2e + 4e) + (-3f - 7f) = 6e - 10f M1 A1
Answer (a): 6e - 10f
(b)
20x + 15 - 3x + 1 = 17x + 16 M1 A1
Answer (b): 17x + 16
(c)
Group matching letters together. M1
e.g. 5m + 4n - 2m + n = 3m + 5n
Answer: 2p(2 + 3q) A1
e.g. 5m + 4n - 2m + n = 3m + 5n
Answer: 2p(2 + 3q) A1
Answer (c): 2p(2 + 3q)
(a) Expand and simplify 4x(2x + 5) – 3x(2x – 3) (2) (b) Given y⁵ × yn / y⁶ = y¹³, find n (2) (c)(i) Solve 7t – 8 < 2t + 7 (2) (ii) Represent on number line (1)
(a) Expand and simplify 4x(2x + 5) - 3x(2x - 3) [2 marks]
(b) work out the value of n. [2 marks]
n =
(c) Solve the inequality 7t - 8 < 2t + 7 [2 marks]
(d) On the number line below, represent the solution set of the inequality solved in part (c)(i) [1 mark]
Worked Solution & Answer
(a)
8x² + 20x – 6x² + 9x = 2x² + 29x M1 A1
Answer (a): 2x2 + 29x
(b)
5 + n – 6 = 13 => n = 14 M1 A1
Answer (b): n = 14
(c)
5t < 15 => t < 3 M1 A1
Answer (c): t < 3
(d)
Open circle at t = 3 with arrow pointing left
Answer (d): Open circle at t = 3 with arrow pointing left