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Gcf

45x^2y^2z^2-38y^5z^3

3 answers

To find the greatest common factor (GCF) of the expression \( 45x^2y^2z^2 - 38y^5z^3 \), we need to analyze the coefficients and the variables separately.

  1. Coefficients:

    • The coefficients are 45 and 38.
    • The GCF of 45 and 38 is 1, since they do not have any common factors other than 1.

  2. Variables:

    • For \( y \), we have \( y^2 \) and \( y^5 \). The GCF is the lowest power, which is \( y^2 \).
    • For \( z \), we have \( z^2 \) and \( z^3 \). The GCF is also the lowest power, which is \( z^2 \).
    • The variable \( x \) only appears in the first term, so it cannot be included in the GCF.

Now we can combine these GCFs:

\[ \text{GCF} = 1 \cdot y^2 \cdot z^2 = y^2z^2 \]

Therefore, the GCF of the expression \( 45x^2y^2z^2 - 38y^5z^3 \) is:

\[ \boxed{y^2z^2} \]

What would the answer be If I have to factor the whole problem

To factor the expression \( 45x^2y^2z^2 - 38y^5z^3 \) entirely, follow these steps:

  1. Identify the Greatest Common Factor (GCF): As we found earlier, the GCF of the given expression is \( y^2z^2 \).

  2. Factor out the GCF: We divide each term in the expression by the GCF \( y^2z^2 \):

    \[ 45x^2y^2z^2 \div y^2z^2 = 45x^2 \] \[ -38y^5z^3 \div y^2z^2 = -38y^{5-2}z^{3-2} = -38y^3z \]

  3. Rewrite the expression: After factoring out the GCF, we can rewrite the expression as:

    \[ 45x^2y^2z^2 - 38y^5z^3 = y^2z^2(45x^2 - 38y^3z) \]

  4. Final factored form: Therefore, the fully factored expression is:

    \[ \boxed{y^2z^2(45x^2 - 38y^3z)} \]

This gives you the expression factored completely, with the common factor outside the parentheses.