# MATH FORMULAS - SPECIAL PRODUCTS, PROOFS, LINKS TO OTHER FORMULAS

Math Formulas

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 = a3 - 3a2b + 3ab2 - b3

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 = a3 + 3a2(-b) + 3a(-b)2 + (-b)3
⇒ (a - b)3 = a3 - 3a2b + 3ab2 - b3 = R.H.S. (Proved.)

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### Proof of Math Formula 8 :

To Prove:
(a + b + c)2 = a2 + b2 + c2 + 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 + c2
= a2 + 2ab + b2 + 2(ac + bc) + c2
= a2 + b2 + c2 + 2ab + 2bc + 2ca = R.H.S. (Proved.)

## Proof of Math Formula 9 :

Proofs of Math Formulas : Formula 9

To Prove:
(a + b + c)(a2 + b2 + c2 - ab - bc - ca) = a3 + b3 + c3 - 3abc

L.H.S. = a(a2 + b2 + c2 - ab - bc - ca) + b(a2 + b2 + c2 - ab - bc - ca)
+ c(a2 + b2 + c2 - ab - bc - ca)
= (a.a2 + a.b2 + a.c2 - a.ab - a.bc - a.ca)
+ (b.a2 + b.b2 + b.c2 - b.ab - b.bc - b.ca)
+ (c.a2 + c.b2 + c.c2 - c.ab - c.bc - c.ca)
= (a3 + ab2 + ac2 - a2b - abc - a2c)
+ (ba2 + b3 + bc2 - ab2 - b2c - abc) + (ca2 + cb2 + c3 - abc - bc2 - c2a)
= (a3 + b3 + c3 - abc - abc - abc)
+ (ab2 - ab2)
+ (ac2 - c2a)
+ ( - a2c + ca2)
Cancelling the same terms with opposite signs, we get
L.H.S. = (a3 + b3 + c3 - 3abc) = R.H.S. (Proved.)

### Proof of Math Formula 10 :

Proofs of Math Formulas : Formula 10

To Prove:
(x + a)(x + b) = x2 + x(a + b) + ab

L.H.S. = (x + a)(x + b) = x(x + b) + a(x + b) = x.x + x.b + a.x + a.b
= x2 + x(a + b) + ab = R.H.S. (Proved.)

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### Proof of Math Formula 11 :

To Prove:
(ax + b)(cx + d) = acx2 + x(ad + bc) + bd

L.H.S. = (ax + b)(cx + d) = ax(cx + d) + b(cx + d)
= ax.cx + ax.d + b.cx + b.d
= acx2 + x(ad + bc) + bd = R.H.S. (Proved.)

### Proof of Math Formula 12 :

To Prove:
(x + a)(x + b)(x + c) = x3 + x2(a + b + c) + x(ab + bc + ca) + abc

L.H.S. = (x + a)(x + b)(x + c) = (x + a){(x + b)(x + c)}
Applying Formula 10 here, we get
L.H.S. = (x + a){x2 + x(b + c) + bc}
= x{x2 + x(b + c) + bc} + a{x2 + x(b + c) + bc}
= x.x2 + x.x(b + c) + x.bc + a.x2 + a.x(b + c) + a.bc
= x3 + x2(b + c) + xbc + ax2 + x(ab + ac) + a.bc
= x3 + x2(a + b + c) + x(ab + bc + ca) + abc = R.H.S. (Proved.)

Thus all the 12 Math Formulas are proved.

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## Solved Examples and Exercise problems on Math Formulas :

Here are the Links for the
Solved Examples and Exercise problems
on application of these Math Formulas.

Set 1 of Solved Examples and Exercise problems

Set 2 of Solved Examples and Exercise problems

Set 3 of Solved Examples and Exercise problems

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