Question

find the gcf (greatest common factor) of the following terms.

\\\\{50x^3, 40x^5\\}\\

Explanation:

Identify the given terms

We are given the set of monomial terms:

$$ \{50x^3, 40x^5\} $$

We need to find their Greatest Common Factor (GCF).

Find the GCF of the numerical coefficients

Using the Greatest Common Factor concept, we find the GCF of the coefficients \(50\) and \(40\).

• Prime factorization of \(50\):

$$ 50 = 2 \times 5^2 $$

• Prime factorization of \(40\):

$$ 40 = 2^3 \times 5 $$

• The common prime factors are \(2\) and \(5\). Taking the lowest power of each:

$$ \text{GCF}(50, 40) = 2^1 \times 5^1 = 10 $$

Find the GCF of the variable parts

Using the Factoring Monomials concept, we find the GCF of the variable parts \(x^3\) and \(x^5\).

• The variable \(x\) is common to both terms.
• We take the lowest exponent of the common variable:

$$ \text{GCF}(x^3, x^5) = x^{\min(3, 5)} = x^3 $$

Combine the numerical and variable GCFs

We multiply the numerical GCF by the variable GCF to get the final greatest common factor:

$$ \text{GCF}(50x^3, 40x^5) = 10 \times x^3 = 10x^3 $$

Answer:

Find the GCF (greatest common factor) of the following terms.

$$ \{50x^3, 40x^5\} $$

Answer: <blank>\(10x^3\)</blank>