Algebra Demystified 2nd Ed

154 algebra De mystif ieD

11. (x + 8)^2 = (x + 8)(x + 8) = x(x) + 8x + 8x + 8(8)
    = x^2 + 16x + 64


12. (x − y)^2 = (x − y)(x − y) = x(x) + x(−y) + x(−y) + (−y)(−y)
    = x^2 − xy − xy + y^2 = x^2 − 2xy + y^2


13. (2x + 3y)^2 = (2x + 3y)(2x + 3y)
    = 2x(2x) + 2x(3y) + 3y(2x) + (3y)(3y)
    = 4x^2 + 6xy + 6xy + 9y^2 = 4x^2 + 12xy + 9y^2


14. xy xyxy
    xx xy xy

+  =  +  + 

=  +  +  + 

()()()

() () ()

2

yyy

xxyy

xxyy

()

= ()+  + ()

=  +  + 

(^22)
2
  2

15. xyxyxx xy xy yy
    xx

+  –  +  –+ +

+ 

()()= () () ()

=()

2

yyx –  yy+ ()=xy – 

2

Factoring Quadratic Polynomials

We now work in the opposite direction with the distributive property—

factoring. First we will factor quadratic polynomials, expressions of the form

ax^2 + bx + c (where a is not 0). For example, x^2 + 5x + 6 is factored as (x + 2)

(x + 3). Quadratic polynomials whose first factors are x^2 are the easiest to factor.

Their factorization always begins as (x ± __ )(x ± __ ). This forces the first term

to be x^2 when the FOIL method is used. All we need to do is to fill in the two

blanks and to decide when to use plus and minus signs. All quadratic polynomi-

als factor, though some do not factor “nicely.” We will only concern ourselves

with “nicely” factorable polynomials in this chapter.

If the second sign is minus, then the signs in the factors will be different (one

plus and one minus). If the second sign is plus then both of the signs will be the

same. If the first sign in the quadratic polynomial is a plus sign, both signs in

the factors will be plus. If the first sign is a minus sign (and the second is a plus

sign), both signs in the factors will be minus.