Questions: Find the greatest common factor of the following list of terms. 6 x^4 y^3, 9 x^5 y^2, and 15 x^3 y^6 The greatest common factor is

Transcript text: Find the greatest common factor of the following list of terms. \[ 6 x^{4} y^{3}, 9 x^{5} y^{2} \text {, and } 15 x^{3} y^{6} \] The greatest common factor is $\square$

Solution

Solution Steps

To find the greatest common factor (GCF) of the given terms, we need to:

Identify the coefficients of each term and find their GCF.
Identify the variables and their exponents in each term and find the minimum exponent for each variable.
Combine the GCF of the coefficients with the variables raised to their minimum exponents.

Step 1: Identify the Coefficients

The coefficients of the given terms are $6$, $9$, and $15$. We need to find the greatest common factor (GCF) of these coefficients.

Step 2: Calculate the GCF of the Coefficients

To find the GCF of $6$, $9$, and $15$: \[ \text{GCF}(6, 9, 15) = 3 \]

Step 3: Identify the Exponents of Variables

The exponents of $x$ in the terms $6x^4y^3$, $9x^5y^2$, and $15x^3y^6$ are $4$, $5$, and $3$, respectively. The exponents of $y$ are $3$, $2$, and $6$.

Step 4: Calculate the Minimum Exponents

The minimum exponent for $x$ is: \[ \min(4, 5, 3) = 3 \] The minimum exponent for $y$ is: \[ \min(3, 2, 6) = 2 \]

Step 5: Construct the GCF Term

Combining the GCF of the coefficients with the variables raised to their minimum exponents, we have: \[ \text{GCF} = 3x^3y^2 \]