To determine if Isiah is correct that 5a² is the greatest common factor (GCF) of the polynomial a³ + 25a²b⁵ + 35b⁴, we need to examine each term of the polynomial and identify the common factors.
The polynomial consists of three terms:
- a³
- 25a²b⁵
- 35b⁴
Now, let’s break down each term to find their factors:
- a³: The factors are a multiplied by itself three times.
- 25a²b⁵: The factors are 5 × 5 × a × a × b × b × b × b × b.
- 35b⁴: The factors are 5 × 7 × b × b × b × b.
Next, we identify the common factors from all three terms:
- For the numerical coefficients: The common factor of 25 and 35 is 5.
- For the variable a: The lowest power of a present is a² (found in 25a²b⁵).
- For the variable b: The lowest power of b present is b^0, as b is not present in the term a³.
Therefore, the GCF of the polynomial a³ + 25a²b⁵ + 35b⁴ is the product of these common factors:
- GCF = 5 × a² = 5a²
Isiah’s determination that 5a² is the GCF of the polynomial is indeed correct!
In summary, by analyzing each term, we found that the greatest common factor that is shared across all terms of the polynomial is 5a².