Question

isiah determined that $5a^2$ is the gcf of the polynomial $a^3 - 25a^2b^5 - 35b^4$. is he correct? explain.

Explanation:

Step1: Analyze coefficients' GCF

Find GCF of 1, 25, 35: $\gcd(1,25,35)=1$

Step2: Analyze variable $a$ terms

Terms: $a^3$, $a^2$, $a^0$ (no $a$ in $-35b^4$). Lowest exponent: $a^0=1$

Step3: Analyze variable $b$ terms

Terms: $b^0$ (no $b$ in $a^3$), $b^5$, $b^4$. Lowest exponent: $b^0=1$

Step4: Combine for overall GCF

Multiply results: $1 \times 1 \times 1 = 1$

Answer:

Isiah is not correct. The GCF of the polynomial $a^3 - 25a^2b^5 - 35b^4$ is 1, because:

1. The greatest common factor of the coefficients 1, 25, and 35 is 1.
2. The term $-35b^4$ has no factor of $a$, so $a^2$ cannot be part of the GCF.
3. The term $a^3$ has no factor of $b$, so there is no common $b$ factor across all terms.