# EXPANSION OF TRINOMIAL WITH POWER 2

Expansion of Trinomial with Power 2 :

In this section, you will learn how to expand a trinomial with power 2.

## Expansion of (a + b + c) Whole Square

(a + b + c)2  =  (a + b + c)(a + b + c)

(a + b + c)2  =  a2 + ab + ac + ab + b+ bc + ac + bc + c2

(a + b + c)2  =  a2 + b+ c+ 2ab + 2bc + 2ac

## Expansion of (a + b - c) Whole Square

To get expansion for (a + b - c)2, let us consider the expansion of (a + b + c)2

(a + b + c)2  =  a2 + b2 + c2 + 2ab + 2bc + 2ac

In (a + b + c)2, if c is negative, then we have

(a + b - c)2

In the terms of the expansion for (a + b + c)2, consider the terms in which we find 'c'.

They are c2, bc, ca.

Even if we take negative sign for 'c' in c2, the sign of c2 will be positive.  Because it has even power 2.

The terms bc, ac will be negative. Because both 'b' and 'a' are multiplied by 'c' that is negative.

Finally, we have

(a + b - c)2  =  a2 + b2 + c2 + 2ab - 2bc - 2ac

## Expansion of (a - b + c) Whole Square

To get expansion for (a - b + c)2, let us consider the expansion of (a + b + c)2

The expansion of (a + b + c)is

(a + b + c)2  =  a2 + b2 + c2 + 2ab + 2bc + 2ca

In (a + b + c)2, if b is negative, then we have

(a - b + c)2

In the terms of the expansion for (a + b + c)2, consider the terms in which we find "b".

They are b2, ab, bc.

Even if we take negative sign for 'b' in b2, the sign of b2 will be positive.  Because it has even power 2.

The terms ab, bc will be negative. Because both 'a' and 'c' are multiplied by 'b' that is negative.

Finally, we have

(a - b + c)2  =  a2 + b2 + c2 - 2ab - 2bc + 2ac

## Expansion of (a - b - c) Whole Square

To get the expansion of (a - b - c)2, let us consider the expansion of (a + b + c)2

The expansion of (a + b + c)is

(a + b + c)2  =  a2 + b2 + c2 + 2ab + 2bc + 2ca

In (a + b + c)2, if b and c are negative, then we have

(a - b - c)2

In the terms of the expansion for (a + b + c)2, consider the terms in which we find 'b' and 'c'.

They are b2, c2, ab, bc, ac.

Even if we take negative sign for 'b' in b2 and negative sign for 'c' in c2, the sign of both band c2 will be positive.  Because they have even power 2.

The terms 'ab' and 'ac' will be negative.

Because, in 'ab', 'a' is multiplied by "b" that is negative.

Because, in 'ac', 'a' is multiplied by "c" that is negative.

The term 'bc' will be positive.

Because, in 'bc', both 'b' and 'c' are negative.

That is,

negative  negative  =  positive

Finally, we have

(a - b - c)2  =  a2 + b2 + c2 - 2ab + 2bc - 2ac

## Expansion of Trinomial with Power 2 - Summary

(a + b + c)2  =  a2 + b2 + c2 + 2ab + 2bc + 2ca

(a + b - c)2  =  a2 + b2 + c2 + 2ab - 2bc - 2ca

(a - b + c)2  =  a2 + b2 + c2 - 2ab - 2bc + 2ca

(a - b - c)2  =  a2 + b2 + c2 - 2ab + 2bc - 2ca

Instead of memorizing all the above formulas, we may memorize the first formula and we may apply values of b and c along with signs.

## Expansion of Trinomial with Power 2 - Questions

Question 1 :

(i) Expand following :

(2x + 3y + 4z)2

Solution :

(a + b + c)2  =  a2 + b2 + c2 + 2ab + 2bc + 2ca

a = 2x, b = 3y and c = 4z

=  (2x)2+(3y)2+(4z)2+2(2x)(3y)+2(3y)(4z)+2(4z)(2x)

=  4x2 + 9y2 + 16z2 + 12xy + 24yz + 16zx

(ii) Expand the following :

(−p +2q + 3r)2

Solution :

a  =  -p, b = 2q and c = 3r

=  (-p)2+(2q)2+(3r)2+2(-p)(2q)+2(2q)(3r)+2(3r)(-p)

=  p2 + 4q2 + 9r2 - 4pq + 12qr - 6rp

## Identity - Product of Three Binomials

(x+a) (x+b) (x+c)

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

(iii) Expand following :

(2p + 3)(2p −4)(2p −5)

Solution :

x  =  2p, a  =  3, b = -4 and c  =  -5

a + b + c  =  3 + (-4) + (-5)  =  -6

ab + bc + ca  =  3(-4) + (-4)(-5) + (-5)(3)

=  -12 + 20 - 15

=  -27 + 20

=  -7

abc  =  3(-4)(-5)  =  60

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

=  (2p)3 + (-6) (2p)2 + (-7)(2p) + 60

=  8p3 -24p2 -14p + 60

(iv) Expand following :

(3a +1)(3a −2)(3a + 4)

Solution :

x  =  3a, a  =  1, b = -2 and c  =  4

a + b + c  =  1 + (-2) + 4  =  3

ab + bc + ca  =  1(-2) + (-2)(4) + 4(1)

=  -2 - 8 + 4

=  -10 + 4

=  -6

abc  =  1(-2)(4)  =  -8

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

=  (3a)3 + 3(3a)2 + (-6)(3a) - 8

=  27a3 + 27a2 -18a - 8 After having gone through the stuff given above, we hope that the students would have understood how to expand a trinomial with power 2.

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 