Products of linear factors

Using the Fundamental Theorem of Algebra, we know that it is possible to express any equation of degree n with one variable as a product of linear factors.

In other words:

xⁿ + a₁x^n-1 + ... + a_n = (x - r₁)(x - r₂)*....*(x - r_n) where r_i represent the n roots for the equation.

In today's, blog, I show that we are not limited to this form.

Lemma 1:

if (r_i ≠ 0), then:

(x - r_i) = 0 if and only if (1 - x/r_i) = 0

Proof

(1) Assume that:

(x - r_i) = 0

(2) Then dividing both sides by (-r_i) gives us:

(1 - x/r_i) = 0

(3) Assume that:

(1 - x/r_i) = 0

(4) Multiply both sides by (-r_i) so that:

(x - r_i) = 0

QED

Corollary 1.1:

if (r_i² ≠ 0), then:

(x² - r_i²) = 0 if and only if (1 - x²/r_i²) = 0

Proof:

This follows directly from Lemma 1 above if set x' = x² and r' = r_i² then we have:

(x² - r_i²) = (1 - x²/r_i²) if and only if (x' - r') = (1 - x'/r')

QED