Questions: Find the GCF for the list. 45 y^3, 63 y^2

Transcript text: Find the GCF for the list. \[ 45 y^{3}, 63 y^{2} \]

Solution

Solution Steps

Step 1: Identify the Polynomials

We are given the polynomials \( 45 y^{3} \) and \( 63 y^{2} \).

Step 2: Factor the Coefficients

First, we need to find the greatest common factor (GCF) of the coefficients \( 45 \) and \( 63 \). The prime factorizations are:

\( 45 = 3^2 \times 5 \)
\( 63 = 3^2 \times 7 \)

The GCF of the coefficients is obtained by taking the lowest power of each common prime factor: \[ \text{GCF}(45, 63) = 3^2 = 9 \]

Step 3: Factor the Variable Parts

Next, we consider the variable parts \( y^{3} \) and \( y^{2} \). The GCF of the variable parts is determined by taking the lowest exponent: \[ \text{GCF}(y^{3}, y^{2}) = y^{\min(3, 2)} = y^{2} \]

Step 4: Combine the GCFs

Finally, we combine the GCF of the coefficients and the GCF of the variable parts to find the overall GCF of the polynomials: \[ \text{GCF}(45 y^{3}, 63 y^{2}) = 9 \cdot y^{2} = 9 y^{2} \]