Polynomials & identities · Ages 13–14
The sum and difference of two cubes
First bracket the same sign, the middle the opposite, the last added.
How to do it
Method
First bracket the same sign, the middle the opposite, the last added
(x + 2)(x2 − 2x + 4) = x3 + 8
Another way
Every term by every term
x3 − 2x2 + 4x + 2x2 − 4x + 8 = x3 + 8
In full
Two identities: a3 + b3 ≡ (a + b)(a2 − ab + b2) and a3 − b3 ≡ (a − b)(a2 + ab + b2). For the signs: the first bracket takes the same sign as the expression; the middle term ab of the second takes the opposite sign; the last term, b2, is always added. Multiply every term by every term to see why: (a + b)(a2 − ab + b2) = a3 − a2b + ab2 + a2b − ab2 + b3, and the four middle terms cancel in pairs, leaving a3 + b3. Spot the cubes first: 8 = 23, 27 = 33, 64 = 43, 125 = 53, and 8x3 = (2x)3. Careful: a sum of cubes is not the cube of a sum; (a + b)3 = a3 + 3a2b + 3ab2 + b3 has two middle terms more. It makes quick sums too: 73 − 33 = (7 − 3)(49 + 21 + 9) = 4 × 79 = 316. Or work it out directly, 343 − 27 = 316: the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
(4x + 1)(16x2 − 4x + 1) = x3 +
Fill in each blank
Show the answer and working
Answer: 64, 1
With a = 4x and b = 1, the second bracket is exactly a2 − ab + b2, so the product is (4x)3 + 13 = 64x3 + 1.
Example 2
64x3 − 27y3 = (x)3 − (y)3
Fill in each blank
Show the answer and working
Answer: 4, 3
64 = 4 × 4 × 4 = 43 and 27 = 33, so 64x3 = (4x)3 and 27y3 = (3y)3.
Example 3
63 + 43 is divisible by 10.
True or false?
Show the answer and working
Answer: True
63 + 43 = (6 + 4)(62 − 6 × 4 + 42), so it is always divisible by 10; and 63 + 43 = 280.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.(x + 7)(x2 − 7x + 49) = x3 +
Answer:
2.Expand (x − 4)(x2 + 4x + 16). What do you get?
- A. x3 + 64
- B. x3 − 64
- C. x3 − 16
- D. x3 − 12x2 + 48x − 64
Choose the right answer
Answer:
3.(x + 9)(x2 + 9x + 81) = x3 + 729 holds for every x.
True or false?
Answer:
4.103 + 53 is divisible by 16.
True or false?
Answer:
5.(x + 7)3 − (x3 + 343) = x2 + x
Fill in each blank
Answer:
All the answers
- 1. 343
- 2. B. x3 − 64
- 3. False
- 4. False
- 5. 21, 147
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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