What is the greatest common factor (GCF) of the polynomial terms 44x⁶ and 8x⁵?

To find the greatest common factor (GCF) of the polynomial terms 44x⁶ and 8x⁵, we need to find the GCF of the numerical coefficients and the GCF of the variable parts.

Step 1: Finding the GCF of the Numerical Coefficients

The numerical coefficients we have are 44 and 8. To find the GCF of these two numbers, we can list their factors or use the prime factorization method.

• 44: The prime factorization of 44 is 2 × 2 × 11 or 2² × 11.
• 8: The prime factorization of 8 is 2 × 2 × 2 or 2³.

The common prime factors between 44 and 8 are 2. The lowest power of 2 that appears in both factorizations is 2² (from 44).

Thus, the GCF of the numerical coefficients is 4.

Step 2: Finding the GCF of the Variable Parts

Next, we need to find the GCF of the variable parts, which are x⁶ and x⁵. To do this, we take the variable with the lowest exponent:

• The lowest exponent of x is 5 (from x⁵).

So, the GCF of the variable parts is x⁵.

Step 3: Combining the GCFs

Now that we have both the GCF of the numerical coefficients and the GCF of the variable parts, we can combine them:

The GCF of 44x⁶ and 8x⁵ is:

4x⁵

Conclusion

Therefore, the greatest common factor of the terms of the polynomial 44x⁶ + 8x⁵ is 4x⁵.