if you have not already done so. There we have listed out all the Algebra Formulas that need to be remembered.

Also, First, study the proofs of First 6 Formulas,
if you have not already done so.

Here we present the proofs of the last six Formulas and provide Links for Solved Examples and Exercise problems on application of those Formulas.

Proofs of 7 to 12 Math Formulas :

Proof of Math Formula 7 :

To Prove:
(a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}

Here we can prove as in Formula 6 above.
(I leave this as exercise for the reader.)

or, we can make use of Formula 6 to prove this.

Replacing b in Formula 6 by (-b), we get
{a + (-b)}^{3} = a^{3} + 3a^{2}(-b) + 3a(-b)^{2} + (-b)^{3}
⇒ (a - b)^{3} = a^{3} - 3a^{2}b + 3ab^{2} - b^{3} = R.H.S. (Proved.)

Get The Best Grades With the Least Amount of Effort : Math Formulas

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To Prove:
(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

L.H.S. = (a + b + c)^{2} = { (a + b) + c }^{2}
Applying Formula1 here,
treating (a + b) as first term and c as second term, we get
L.H.S. = (a + b)^{2} + 2(a + b)c + c^{2}
= a^{2} + 2ab + b^{2} + 2(ac + bc) + c^{2}
= a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca = R.H.S. (Proved.)

Proof of Math Formula 9 :

Proofs of Math Formulas : Formula 9

To Prove: (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca) = a^{3} + b^{3} + c^{3} - 3abc

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