D'Alembert's Theorem

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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:

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.

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Example 2

Check that the polynomial P(x) = 12x³ + 4x² – 8x is divisible by 4x.

1st) We determine the root of 4x:

-b/a = -0/4 = 0

2nd) We replace x by 0 in the polynomial P(x) = 12x³ + 4x² – 8x:

P(0) = 12.0³ + 4.0² - 8.0
P(0) = 0 + 0 - 0
P(0) = 0

Since P(0) = 0, the polynomial P(x) = 12x³ + 4x² – 8x is divisible by 4x.

Example 3

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

1st) We determine the root of x – 2:

-b/a = -(-2)/1 = 2

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

P(2) = 2² - 2.2 + 1
P(2) = 4 - 4 +1
P(2) = 1

Since P(2) ≠ 0, the polynomial P(x) = x² – 2x + 1 is not divisible by x – 2.

You may also be interested:

• Polynomial Division - Key Method
• polynomial function
• Polynomial Factoring

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