Question

Find the GCD of the following $(a-1{)}^5(a+3{)}^2,(a-2{)}^2(a-1{)}^3(a+3{)}^4$

• A. $(a-1{)}^3(a+3{)}^2$
• B. $(a+1{)}^3(a+3{)}^2$
• C. $(a-1{)}^3(a-3{)}^2$
• D. None of these

Answer 1

The correct option is A $(a-1{)}^3(a+3{)}^2$

We first factorize the given polynomials $(a-1{)}^5(a+3{)}^2$ and $(a-2{)}^2(a-1{)}^3(a+3{)}^4$ as shown below:

$(a-1{)}^5(a+3{)}^2=(a-1)\cdot (a-1)\cdot (a-1)\cdot (a-1)\cdot (a-1)\cdot (a+3)\cdot (a+3)\cdot (a+3)$

$(a-2{)}^2(a-1{)}^3(a+3{)}^4=(a-2)\cdot (a-2)\cdot (a-1)\cdot (a-1)\cdot (a-1)\cdot (a+3)\cdot (a+3)\cdot (a+3)\cdot (a+3)$

The common factors of $(a-1{)}^5(a+3{)}^2$ and $(a-2{)}^2(a-1{)}^3(a+3{)}^4$ are $(a-1{)}^3$ and $(a+3{)}^2$, therefore, the GCD is $(a-1{)}^3(a+3{)}^2$.

Hence, the greatest common divisor is $(a-1{)}^3(a+3{)}^2$.