D'Alembert's Theorem

protection click fraud

O D'Alembert's theorem is lets know if a polynomialP(x) is divisible by a binomial of type ax + b, even before performing the division between them.

In other words, the theorem allows us to know whether the remainder R of the division is equal to zero or not. This theorem is an immediate consequence of the rest theorem for division of polynomials. Understand why below.

rest theorem

When dividing a polynomial P(x) by a binomial of type ax + b, the remainder R is equal to the value of P(x) when x is the root of the binomial ax + b.

Root of the binomial: ax + b = 0 ⇒ x = -b/a. So, by the rest theorem, we have to:

R = P(-b/a)

Now, see that if P(-b/a) = 0, then R = 0 and if R = 0, we have divisibility between the polynomials. And that's exactly what D'Alembert's theorem tells us.

D'Alembert's Theorem: if P(-b/a) = 0, then the polynomial P(x) is divisible by the binomial ax + b.

Example 1

Check that the polynomial P(x) = 6x² + 2x is divisible by 3x + 1.

1st) We determine the root of 3x + 1:

-b/a = -1/3

2) We replace x by -1/3 in the polynomial P(x) = 6x² + 2x:

instagram story viewer

P(-1/3) = 6.(-1/3)² + 2.(-1/3)
P(-1/3) = 6.(1/9) + 2.(-1/3)
P(-1/3) = 6/9 - 2/3
P(-1/3) = 2/3 - 2/3
P(-1/3) = 0

Since P(-1/3) = 0, the polynomial P(x) = 6x² + 2x is divisible by 3x + 1.

Check out some free courses