Prove that (n – 2)to the power of 3 + 6 (n – 2) to the power of 2 + 12 (n – 2) +8 = n to the power of 3
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2007-06-26 06:29:00 · 7 answers · asked by rma 1 in Science & Mathematics ➔ Mathematics
(n - 2)^3 + 6(n - 2)^2 + 12(n - 2) + 8
(n^2 - 4n + 4)(n - 2) + 6n^2 - 24n + 24 + 12n - 24 + 8
n^3 - 6n^2 + 12n - 8 + 6n^2 -12n + 8
n^3
2007-06-26 06:37:42 · answer #1 · answered by yeeeehaw 5 · 2⤊ 1⤋
To prove this kind of equation, you expanse and then simplify the LHS, if it equals the RHS then the proof is done. So you should proceed as follow:
LHS = (n-2)^3 +6(n-2)^2 + 12(n-2) +8
= n^3-6n^2 +12n -8 +6n^2 -24n +24+12n-24 + 8
= n^3
= RHS
Q.E.D.
2007-06-27 01:30:00 · answer #2 · answered by ddntruong 2 · 0⤊ 0⤋
I will complete this proof by transforming the left hand side of the equation to the right hand side of the equation.
(n - 2)^3 + 6(n - 2)^2 + 12(n - 2) + 8
= (n - 2)(n - 2)(n - 2) + 6(n - 2)(n - 2) + 12(n - 2) + 8
= (n^2 - 4n + 4)(n - 2) + 6(n^2 - 4n + 4) + 12n - 24 + 8
= n^3 - 4n^2 + 4n - 2n^2 + 8n - 8 + 6n^2 - 24n + 24 +12n - 24 + 8
= n^3 + 6n^2 - 6n^2 + 24n - 24n + 32 - 32
= n^3
Therefore, (n - 2)^3 + 6(n - 2)^2 + 12(n - 2) + 8 = n^3.
2007-06-26 06:59:38 · answer #3 · answered by mathjoe 3 · 0⤊ 1⤋
Prove that (n – 2)to the power of 3 + 6 (n – 2) to the power of 2 + 12 (n – 2) +8 = n to the power of 3
(n-2)^3 +6(n-2)^2 + 12(n-2) +8 = n^3
n^3-6n^2 +12n -8 +6n^2 -24n +24+12n-24 + 8 = n^3
n^3=n^3
2007-06-26 06:44:05 · answer #4 · answered by ironduke8159 7 · 0⤊ 1⤋
Rewriting:
(n-2)^3 + 6(n-2)^2 + 12(n-2) + 8
This can be done with brute force:
(n-2)^3 = n^3 - 6n^2 + 12n - 8 by foiling twice
6(n-2)^2 = 6(n^2 - 4n + 4) = 6n^2 - 24n + 24
12(n-2) = 12n - 24
8 = 8
Add'em all up:
n^3 + (6n^2-6n^2) + (12n - 24n +12n) + (-8 + 24 - 24 + 8), which simplifies to n^3.
I'm sure there is a more elegant solution by substituting m = n-2, but this one works OK.
2007-06-26 06:47:56 · answer #5 · answered by MathProf 4 · 0⤊ 1⤋
Ist part (n-2)^3 = (n^2 - 4n +4)(n - 2) = n^3 - 6n^2 + 12n - 8
2nd part 6(n - 2)^2 = 6(n^2 - 4n +4) = 6n^2 - 24n +24
3rd part 12(n-2) +8 = 12n - 24 +8 = 12n - 16
Added together:-
n^3 - 6n^2 +12n -8 + 6n^2 - 24n +24 +12n - 16
Everything cancels out leaving n^3
2007-06-26 08:31:30 · answer #6 · answered by brainyandy 6 · 1⤊ 1⤋
(n - 2)^3 + 6 (n - 2)^2 + 12 (n - 2) + 8
=(n - 2)^3 + 6 (n^2 - 4n + 4) + 12n - 24 + 8
=(n - 2)^3 + 6n^2 - 24n + 24 + 12n - 16
=(n - 2)^3 + 6n^2 - 12n + 8
=(n - 2)(n^2 - 4n + 4) + 6n^2 - 12n + 8
=n^3 - 4n^2 + 4n - 2n^2 + 8n - 8 + 6n^2 - 12n + 8
=n^3
2007-06-26 07:15:02 · answer #7 · answered by fofo m 3 · 1⤊ 0⤋