D'Alembert's theorem is an immediate consequence of the remainder theorem, which are concerned with the division of polynomial by binomial of type x – a. The remainder theorem says that a polynomial G(x) divided by a binomial x – a will have remainder R equal to P(a), for
x = a. The French mathematician D'Alembert proved, taking into account the theorem cited above, that a polynomial any Q(x) will be divisible by x – a, that is, the remainder of the division will be equal to zero (R = 0) if P(a) = 0.
This theorem made it easier to calculate the division of polynomial by binomial (x –a), thus it was not necessary to solve the entire division to know if the remainder is equal to or different from zero.
Example 1
Calculate the remainder of the division (x2 + 3x – 10): (x – 3).
As D'Alembert's Theorem says, the remainder (R) of this division will be equal to:
P(3) = R
32 + 3 * 3 – 10 = R
9 + 9 - 10 = R
18 - 10 = R
R = 8
So the rest of this division will be 8.
Example 2
Check if x5 – 2x4 + x3 + x – 2 is divisible by x – 1.
According to D’Alembert, a polynomial is divisible by a binomial if P(a) = 0.
P(1) = (1)5 – 2*(1)4 + (1)3 + (1) – 2
P(1) = 1 - 2 + 1 + 1 - 2
P(1) = 3 - 4
P(1) = – 1
Since P(1) is non-zero, the polynomial will not be divisible by the binomial x – 1.
Example 3
Calculate the value of m so that the remainder of the division of the polynomial
P(x) = x4 – mx3 + 5x2 + x – 3 by x – 2 is 6.
We have that, R = P(x) → R = P(2) → P(2) = 6
P(2) = 24 – m*23 + 5*22 + 2 – 3
24 – m*23 + 5*22 + 2 – 3 = 6
16 – 8m + 20 + 2 – 3 = 6
– 8m = 6 – 38 + 3
– 8m = 9 – 38
– 8m = – 29
m = 29/8
Example 4
Calculate the remainder of the division of the 3x polynomial3 + x2 – 6x + 7 by 2x + 1.
R = P(x) → R = P(– 1/2)
R = 3*(–1/2)3 + (–1/2)2 – 6*(–1/2) + 7
R = 3*(–1/8) + 1/4 + 3 + 7
R = –3/8 + 1/4 + 10 (mmc)
R = –3/8 + 2/8 + 80/8
R = 79/8
by Mark Noah
Graduated in Mathematics
Brazil School Team
Polynomials - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/teorema-dalembert.htm