Verified answer

Step-by-step explanation:

Let's factorize each expression using suitable identities:

(a) \(x^2 - 6x + 9\)

This expression is a perfect square trinomial in the form \(a^2 - 2ab + b^2\), where \(a = x\) and \(b = 3\). We can use the identity \((a - b)^2 = a^2 - 2ab + b^2\) to factorize it:

\(x^2 - 6x + 9 = (x - 3)^2\)

(b) \(p^2 - 18p + 81\)

This expression is also a perfect square trinomial, where \(a = p\) and \(b = 9\). We can use the same identity to factorize it:

\(p^2 - 18p + 81 = (p - 9)^2\)

(c) \(m^4 + 2m^2n^2 + n^4\)

This expression is a sum of cubes in the form \(a^3 + 3a^2b + 3ab^2 + b^3\), where \(a = m^2\) and \(b = n^2\). We can use the identity \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) to factorize it:

\(m^4 + 2m^2n^2 + n^4 = (m^2 + n^2)^2\)

(d) \(a^6 + 8a^3 + 16\)

This expression is also a sum of cubes in the form \(a^3 + 2b^3\), where \(a = a^2\) and \(b = 2\). We can use the identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) to factorize it:

\(a^6 + 8a^3 + 16 = (a^2 + 2)(a^4 - 2a^2 + 4)\)

So, the factorized expressions are:

(a) \(x^2 - 6x + 9 = (x - 3)^2\)

(b) \(p^2 - 18p + 81 = (p - 9)^2\)

(c) \(m^4 + 2m^2n^2 + n^4 = (m^2 + n^2)^2\)

(d) \(a^6 + 8a^3 + 16 = (a^2 + 2)(a^4 - 2a^2 + 4)\)