BASIC ALGEBRA FORMULAS - SPECIAL PRODUCTS, PROOFS, LINKS TO OTHER FORMULAS

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

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

Proofs of Basic Algebra Formulas in Polynomials :

Proof of Basic Algebra Formula 1 :

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

L.H.S. = (a + b)^{2} = (a + b)(a + b) = a(a + b) + b(a + b)
= a.a + a.b + b.a + b.b = a^{2} + (ab + ab) + b^{2}
= a^{2} + 2ab + b^{2} = R.H.S. (Proved.)

Proof of Basic Algebra Formula 2 :

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

Here we can directly prove as in Formula 1 above.
(I leave this as exercise for the reader.)

or, we can make use of Formula 1 to prove this.

Replacing b in Formula 1 by (-b), we get
{a + (-b)}^{2} = a^{2} + 2a(-b) + (-b)^{2}
⇒ (a - b)^{2} = a^{2} - 2ab + b^{2} (Proved.)

Proof of Basic Algebra Formula 3 :

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

L.H.S. = (a + b)(a - b)
= a(a - b) + b(a - b)
= a.a - a.b + b.a - b.b = a^{2} + (-ab + ab) - b^{2} = a^{2} - b^{2} = R.H.S. (Proved.)

Proof of Basic Algebra Formula 4 :

Proving one of the Basic Algebra Formulas :

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

L.H.S. = (a + b)(a^{2} - ab + b^{2}) = a(a^{2} - ab + b^{2}) + b(a^{2} - ab + b^{2})

= a^{3} + (- a^{2}b + ba^{2}) + (ab^{2} - b^{2}a) + b^{3} The terms in brackets are same and cancel each other. ∴ L.H.S. = a^{3} + b^{3} = R.H.S. (Proved.)

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

Here we can directly prove as in Formula 4 above.
(I leave this as exercise for the reader.)
or, we can make use of Formula 4 to prove this.

Replacing b in Formula 4 by (-b), we get
{a + (-b)}{a^{2} - a(-b) + (-b)^{2}} = a^{3} + (-b)^{3}
⇒ (a - b)(a^{2} + ab + b^{2}) = a^{3} - b^{3} = R.H.S. (Proved.)

Proof of Basic Algebra Formula 6 :

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

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