XIX Multiplication of Expressions

(y+b)(x+a)=y(x+a)+b(x+a)(y+b)(x+a) = y(x+a) + b(x+a)
=xy+ay+bx+ab= xy+ay+bx+ab

((x+y)+1)2=(x+y+1)2=(x+y+1)(x+y+1)((x+y)+1)^2 = (x+y+1)^2 = (x+y+1)(x+y+1)

If the first terms in each factor are alike:

(x+a)(x+b)=x(x+b)+a(x+b)(x+a)(x+b) = x(x+b) + a(x+b)
=x2+bx+ax+ab= x^2+bx+ax+ab
=x2+x(a+b)+ab= x^2+x(a+b)+ab

This still holds true if the coefficients of the first term are not 11:

(px+a)(qx+b)=px(qx+b)+a(qx+b)(px+a)(qx+b) = px(qx+b)+a(qx+b)
=pqx2+pbx+aqx+ab= pqx^2+pbx+aqx+ab
=pqx2+x(bp+aq)+ab= pqx^2+x(bp+aq)+ab

Square of an Expression With Two Terms

(x+a)2=(x+a)(x+a)(x+a)^2 = (x+a)(x+a)
(x+a)2=x2+2ax+a2(x+a)^2 = x^2+2ax+a^2
(xa)2=x22ax+a2(x-a)^2 = x^2-2ax+a^2

Square of an Expression With Three Terms

Consider the rule (x+c)2=x2+2cx+c2(x+c)^2 = x^2+2cx+c^2. As xx may have any value, replace it with (a+b)(a+b):

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


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

The rules for signs in the distributed expression are:
\Rightarrow The individual squares are always positive!
\Rightarrow The double of a combination of two terms is negative when any of its terms is negative!

(xy+z)2=x2+y2+z22xy+2xz2yz(x-y+z)^2 = x^2+y^2+z^2-2xy+2xz-2yz
(abc)2=a2+b2+c22ab2ac2bc(a-b-c)^2 = a^2+b^2+c^2-2ab-2ac-2bc
(x+y5)2=x2+y2+2510x10y+2xy(x+y-5)^2 = x^2+y^2+25-10x-10y+2xy

Cube of an Expression With Two Terms

(a+b)3=(a+b)(a+b)(a+b)(a+b)^3 = (a+b)(a+b)(a+b)
=(a+b)(a2+2ab+b2)= (a+b)(a^2+2ab+b^2)
=a3+2a2b+ab2+a2b+2ab2+b3= a^3+2a^2b+ab^2+a^2b+2ab^2+b^3
=a3+3a2b+3ab2+b3= a^3+3a^2b+3ab^2+b^3

(ab)3=a33a2b+3ab2b3(a-b)^3 = a^3-3a^2b+3ab^2-b^3

Negative Expressions With Exponents

Distribute the expression according to the exponent before applying the negation:

(ab)2=(a22ab+b2)-(a-b)^2 = -(a^2-2ab+b^2)
=a2+2abb2= -a^2+2ab-b^2

Multiplication of Monomial Expressions

(3a)2=(3a)(3a)(3a)^2 = (3a)(3a)
=9a2= 9a^2

Product of Sum and Difference

The product of the sum and the difference of two numbers always reduces to the difference of their squares.

(a+b)(ab)=a(ab)+b(ab)(a+b)(a-b) = a(a-b) + b(a-b)
=a2ab+abb2= a^2-ab+ab-b^2
=a2b2= a^2-b^2


a(b+c)(d+e)=a(bd+be+cd+ce)a(b+c)(d+e) = a(bd+be+cd+ce)
=abd+abe+acd+ace= abd+abe+acd+ace