Find the greatest common factor of a^2+ab,a^2-b^2 and (a+b)^2
You asked:
Find the polynomial greatest common divisor of the following expressions: \(a^{2} + a b\), \(a^{2} - b^{2}\), \(\left(a + b\right)^{2}\)
MathBot Answer:
The greatest common divisor is \(a + b\).
Begin by factoring all expressions completely. $$\begin{aligned}a^{2} + a b &= a \left(a + b\right)\\a^{2} - b^{2} &= \left(a - b\right) \left(a + b\right)\\\left(a + b\right)^{2} &= \left(a + b\right)^{2}\\\end{aligned}$$
Find the greatest common divisor by multiplying common factors together, including only the lowest power of those factors which are common to all expressions. \[\operatorname{gcd}(a^{2} + a b, a^{2} - b^{2}, \left(a + b\right)^{2}) = (a + b)^{1} = a + b\]