Find the GCF of monomials to factorize an expression with monomials Precalculus

One variable is common in all three terms. To find the greatest common factor, we choose the variable with the smallest exponent. That is x^2.

The coefficient is the greatest common divisor of the coefficients. Here, we have:

f\left(x\right)=6x^5+3x^4-3x^2

Therefore, the coefficients are 6, 3 and -3. We can write:

gcd\left(6{,}3,-3\right)= gcd\left(6{,}3{,}3\right) = 3

The greatest common factor is 3x^2.

6x^5+3x^4-3x^2\\ = 3x^2\left(\dfrac{6x^5}{3x^2} + \dfrac{3x^4}{3x^2} - \dfrac{3x^2}{3x^2}\right)\\=3x^2\left(2x^3+x^2-1\right)

The variables x and y are common in all three terms. To find the greatest common factor, we choose the variable with the smallest exponent. That is x and y^2.

The coefficient is the greatest common divisor of the coefficients. Here, the coefficients are 2{,}6 and 4. We can write:

gcd\left(2{,}6{,}4\right)= 2

The greatest common factor is 2xy^2.

Therefore:

f\left(x\right)=2x^3y^2+6xy^3+ 4x^4y^4\\= 2xy^2\left( \dfrac{2x^3y^2}{2xy^2} + \dfrac{6xy^3}{2xy^2} + \dfrac{4x^4y^4}{2xy^2}\right)\\= 2xy^2\left(x^2+3y+ 2x^3y^2\right)

To find the greatest common factor, we choose the variable with the smallest exponent. That is x^5.

The coefficient is the greatest common divisor of the coefficients. Here, we have:

gcd\left(8{,}4\right)= 4

The greatest common factor is 4x^5.

Therefore:

8x^7+4x^5 = 4x^5\left(\dfrac{8x^7}{4x^5} + \dfrac{4x^5}{4x^5}\right) = 4x^5\left(2x^2+1\right)

Two variables are common in all three terms. To find the greatest common factor, we choose the variable with the smallest exponent. That is x and y

The coefficient is the greatest common divisor of the coefficients. Here, all the coefficients are 1.

The greatest common factor is xy.

Therefore:

x^3y + xy^4+xy\\ = xy\left(\dfrac{x^3y}{xy} + \dfrac{xy^4}{xy} +\dfrac{xy}{xy}\right)\\=xy\left(x^2+y^3+1\right)

Only one variable, namely x, is common in both terms. To find the greatest common factor, we choose the variable with the smallest exponent. That is x.

The coefficient is the greatest common divisor of the coefficients. Here, the coefficients are 4 and -8. We can write:

gcd\left(4,-8\right)= gcd\left(4{,}8\right) = 4

The greatest common factor is 4x.

Therefore:

4x^3-8xy\\ = 4x\left(\dfrac{4x^3}{4x} - \dfrac{8xy}{4x}\right)\\=4x\left(x^2-2y\right)

The variables y and z are common in all three terms. To find the greatest common factor, we choose the variable with the smallest exponent. That is y and z.

The greatest common factor is yz.

Therefore:

x ^ 2 {yz} -y ^ 4z + yz \\ = yz \left(\dfrac {x ^ 2 {yz}} {yz} - \dfrac {y ^ 4z} {yz} + \dfrac {yz} {yz} \right) \\ = yz \left(x ^ 2 - y ^ 3 + 1\right)

Hence we have:

To find the greatest common factor, we choose the variables with the smallest exponent. That is x and y.

The coefficient is the greatest common divisor of the coefficients. Here, we have:

gcd\left(2{,}3,-5\right)= gcd\left(2{,}3{,}5\right) = 1

The greatest common factor is xy.

Therefore:

2xy^2+3xy-5x^2y\\ = xy\left(\dfrac{2xy^2}{xy} + \dfrac{3xy}{xy} - \dfrac{5x^2y}{xy}\right)\\=xy\left(2y+3-5y\right)