Factoring Quadratic Expressions

Question

What is the relationship between the Factors and the Factorised Expression?

Answer 1

a^2 - b^2 = (a+b)(a-b)

the factors are a+b and a-b

the factorized expression is (a+b)(a-b)

multiply the factors together to get the factorized expression

(not sure if you're up to more than that or not, in the question)

another approach is take out the common factor

start with a^2 +a^4

factorize it to a^2(1+a^2)

the factorized expression is the product of the factors

similar to the first example above

or do it with more than 2 factors, to complicate things

Answer 2

It is a useful thing

say f(x)=k[xⁿ+a₁x^(n-1)+a₂x^(n-2)+...+a_(n-1)x+a_n]

then say it can be rewritten f(x)=k[(x-r₁)(x-r₂)...(x-r_n)]

it is kind of obvious that a_n=[(-1)ⁿ]r₁r₂...r_n

less obvious is a₁=-(r₁+r₂+...+r₃)

if a₁=0, then the sum of the roots equal zero

for example, since cos(pi/9), cos(7pi/9), cos(13pi/9) are the roots of

f(x)=4x³-3x-1/2=4[(x-cos(pi/9)(x-cos(7pi/9)(x-cos(13pi/9)]

then cos(pi/9)+cos(7pi/9)+cos(13pi/9)=0

and (-1)cos(pi/9)cos(7pi/9)cos(13pi/9)=-1/8