How to find GCD of polynomials

O greatest common divisor(MDC) between two or more whole numbers corresponds to the largest divider common that exists between them. In between polynomials, the MDC has the same idea.

Thus, to understand how to calculate the GCD between polynomials, it is important to know how to calculate the GCD of integers.

In a practical way, the MDC can be obtained as the product of the prime factors common that exist between the numbers.

Example: Calculate GCD between 16 and 24.

Decomposition into prime factors:

The GCD between 16 and 24 is the product of the factors common to the two numbers, that is,

GCD(16, 24) = 2. 2. 2 = 8.

Now let's see how to find GCD of polynomials. We will start with the simplest case, with polynomials formed by a single term: the monomials.

GCD of monomials

Let's see some examples of how to calculate the GCD between two or more monomials.

Example 1: MDC between 6x and 15x.

Decomposing into prime factors, we have:

6 = 2. 3 and 15 = 3. 5

Therefore, we can write each of the monomials as follows:

6x = 2. 3. x

15x = 3. 5. x

Therefore, the MDC is 3x.

Example 2: MDC between 18x²y and 30xy.

Decomposing into prime factors, we have:

18 = 2. 3. 3 and 30 = 2. 3. 5

Therefore, we can write each of the monomials as follows:

18x²y = 2. 3. 3. x². y = 2. 3. 3. x. x. y

30xy = 2. 3. 5. x. y 

2. 3. x. y = 6x

So, the MDC is 6xy.

GCD of polynomials

To find the GCD of polynomials, we first check whether it is possible to factor each of them. For this, we use techniques of polynomial factorization.

Example 1: GCD between (x² – y²) and (2x – 2y).

Note that the first polynomial corresponds to a difference of two squares. So we can factor it as follows:

x² – y² = (x – y).(x + y)

Already in the second polynomial, we can write the common factor, 2, in evidence:

2x – 2y = 2.(x – y)

In this way, we have:

x² – y² = (x – y).(x + y)

2x – 2y = 2.(x – y)

So, the GCD between the polynomials is (x – y).

Example 2: GCD between (x³ + 27) and (x² + 6x + 9).

The first polynomial corresponds to a sum between two cubes, see:

x³ + 27 = x³ + 3³ = (x + 3).(x² – 3x + 9)

And the second polynomial, squared to the sum of two terms:

x² + 6x + 9 = (x + 3)² = (x + 3).(x + 3)

So, we have to:

x³ + 27 = (x + 3).(x² – 3x + 9)

x² + 6x + 9 = (x + 3).(x + 3)

Therefore, the GCD between the polynomials is (x + 3).

Example 3: GCD between (2x² – 32) and (x³ + 12x² + 48x + 64).

Here, the first polynomial is a difference between two squares:

2x² – 32 = 2.(x² – 16) = 2.(x² – 4²) = 2.(x – 4).(x + 4)

Meanwhile, the second polynomial is the cube of the sum of two terms:

x³ + 12x² + 48x + 64 = (x)³ + 3. (x²). (4) + 3. (4²). (x) + (4)³ = (x + 4)³ = (x + 4).(x + 4).(x + 4)

So, we have to:

2x² – 32 = 2.(x – 4).(x + 4)

x³ + 12x² + 48x + 64 = (x + 4).(x + 4).(x + 4)

Therefore, the GCD between the polynomials is (x + 4).

Difference between MMC and MDC

Confusion between the concepts of MDC and MMC (least common multiple). However, while GCD corresponds to the highest common divisor, MMC is given by the lowest common multiple.

MMC is a very useful tool in solving fractional equations because, in general, the denominators of the fractions they're not the same.

In these situations, what we do is extract the MMC between the denominators and from there write equivalent fractions of the same denominator.

However, denominators are not always known numbers, they can be algebraic expressions or polynomials. Therefore, it is common to have to calculate the polynomial MMC.

At this time, it is important not to confuse and want find the GCD of the equation, when what has to be calculated is the MMC of the equation.

You may also be interested:

• Exercises on Equivalent Fractions
• How to add and subtract fractions
• Practical method to solve 1st degree equations
• Divisibility criteria