Find the value of
- (i) x3 + y3 − 12xy + 64, when x + y = −4
- (ii) x3 − 8y3 − 36xy − 216, when x = 2y + 6
Step-by-step solution
Idea: A challenge question. Use the identity a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca). If a + b + c = 0, the right side is 0, so a3 + b3 + c3 − 3abc = 0 (that is, a3 + b3 + c3 = 3abc). Move everything in the given condition to one side to find the third term c.
(i) x3 + y3 − 12xy + 64, when x + y = −4
- Write the condition with 0 on one side: x + y = −4 means x + y + 4 = 0. So take a = x, b = y, c = 4; then a + b + c = 0.
- Check the expression matches: c3 = 43 = 64 and 3abc = 3 × x × y × 4 = 12xy.
So x3 + y3 − 12xy + 64 = x3 + y3 + 43 − 3(x)(y)(4)1 mark - = (x + y + 4)(x2 + y2 + 16 − xy − 4y − 4x) = 0 × (…) = 01 mark
(ii) x3 − 8y3 − 36xy − 216, when x = 2y + 6
- Write the condition with 0 on one side: x = 2y + 6 means x − 2y − 6 = 0, i.e. x + (−2y) + (−6) = 0. So take a = x, b = −2y, c = −6; then a + b + c = 0.
- Check the expression matches: b3 = (−2y)3 = −8y3, c3 = (−6)3 = −216, and 3abc = 3 × x × (−2y) × (−6) = 36xy.
So x3 − 8y3 − 36xy − 216 = x3 + (−2y)3 + (−6)3 − 3(x)(−2y)(−6)1 mark - Since a + b + c = x − 2y − 6 = 0, the identity gives a3 + b3 + c3 − 3abc = 0. The value is 0.1 mark
Check: (i) Take x = 1, y = −5 (so x + y = −4): 1 − 125 + 60 + 64 = 0 ✓. (ii) Take y = 1, x = 8 (so x = 2y + 6): 512 − 8 − 288 − 216 = 0 ✓.
Answer to write in the exam
(i)
x + y = −4 ⇒ x + y + 4 = 0
x3 + y3 − 12xy + 64 = x3 + y3 + 43 − 3(x)(y)(4)
= (x + y + 4)(x2 + y2 + 16 − xy − 4y − 4x) [a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca)]
= 0 × (x2 + y2 + 16 − xy − 4y − 4x)
∴ x3 + y3 − 12xy + 64 = 0
(ii)
x = 2y + 6 ⇒ x + (−2y) + (−6) = 0
x3 − 8y3 − 36xy − 216 = x3 + (−2y)3 + (−6)3 − 3(x)(−2y)(−6)
= (x − 2y − 6)(x2 + 4y2 + 36 + 2xy − 12y + 6x) [a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ca)]
= 0 × (x2 + 4y2 + 36 + 2xy − 12y + 6x)
∴ x3 − 8y3 − 36xy − 216 = 0
Common mistakes that cost marks
- Taking c = −4 in (i) because the condition says −4. Move it across first: x + y + 4 = 0, so c = +4 (and 43 = 64 matches the expression).
- In (ii), taking b = 2y or c = 6. The signs must match the expression: (−2y)3 = −8y3 and (−6)3 = −216.
- Getting the sign of 3abc wrong: 3 × x × (−2y) × (−6) is +36xy (two negatives make a positive), so −3abc = −36xy.
How this can come in the exam
If a + b + c = 0 and abc = 4, then a3 + b3 + c3 is equal to
- 0
- 4
- 12
- 64
Show answer
(C) 12
a3 + b3 + c3 − 3abc = (a + b + c)(…) = 0, so a3 + b3 + c3 = 3abc = 3 × 4 = 12.
Without finding the cubes, find the value of 283 + (−15)3 + (−13)3.
Show answer
Here a = 28, b = −15, c = −13 and a + b + c = 28 − 15 − 13 = 0 (1 mark). So a3 + b3 + c3 = 3abc = 3 × 28 × (−15) × (−13) = 3 × 28 × 195 = 16380 (1 mark).Try one yourself
Find the value of x3 + y3 − 9xy + 27, when x + y = −3.
Show answer
x + y + 3 = 0, so take a = x, b = y, c = 3: 33 = 27 and 3abc = 9xy. The expression is a3 + b3 + c3 − 3abc with a + b + c = 0, so its value is 0.
More questions like this
- Consider any three consecutive square numbers. For example, 1, 4, and 9. Add the smallest and the largest squares. Thus, 1 + 9 = 10. Then subtract twice the middle square from this sum. This leads to 10 − (2 × 4) = 10 − 8 = 2. Now try the same process with another set of three consecutive square numbers. Say 9, 16, 25.
- Try and find other patterns like this one. For example, you could consider 4 consecutive squares and see if you can find a pattern.
- Think of numbers a and b where a and b do not represent lengths of line segments. What if a and b are negative numbers? Let us check for some negative numbers and see if this equation still works.
- What is the difference between an equation and an identity?
- By now you must have observed that (a + b)2 ≠ a2 + b2. But can you find out which one of them will be greater? For example, if a = 10 and b = 2, what are the values of (a + b)2 and a2 + b2?