Find the G.C.D. of x^{2}-4 and x^{3}-5x+6. | vedclass

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(B) Let $p(x) = x^{2}-4$.Using the identity $a^{2}-b^{2} = (a-b)(a+b)$,we get:$p(x) = (x-2)(x+2)$.Now,let $q(x) = x^{2}-5x+6$.To factorize this quadra

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$x-2$

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(B) Let $p(x) = x^{2}-4$.Using the identity $a^{2}-b^{2} = (a-b)(a+b)$,we get:$p(x) = (x-2)(x+2)$.Now,let $q(x) = x^{2}-5x+6$.To factorize this quadratic,we look for two numbers whose product is $6$ and sum is $-5$. These numbers are $-2$ and $-3$.$q(x) = x^{2}-2x-3x+6$$q(x) = x(x-2)-3(x-2)$$q(x) = (x-2)(x-3)$.The $G.C.D.$ (Greatest Common Divisor) is the highest degree common factor present in both polynomials.Comparing $p(x) = (x-2)(x+2)$ and $q(x) = (x-2)(x-3)$,the common factor is $(x-2)$.Therefore,the $G.C.D.$ is $x-2$.

Find the $G.C.D.$ of $x^{2}-4$ and $x^{3}-5x+6$.

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