(2) Hence, prove by means of factorisation, that \( x(x-3)^{2}-3(3-x)^{2}=(x-3)^{3} \)

To prove the identity \[ x(x-3)^{2} - 3(3-x)^{2} = (x-3)^{3}, \] let's proceed step-by-step using factorization. ### Step 1: Recognize a Common Factor Notice that both terms on the left-hand side contain \((x-3)^{2}\). Let's factor this common term out: \[ x(x-3)^{2} - 3(3-x)^{2} = (x-3)^{2} \cdot x - 3(3-x)^{2}. \] ### Step 2: Simplify \((3-x)^{2}\) Observe that \((3-x)^{2}\) is the same as \((x-3)^{2}\) because squaring eliminates the sign: \[ (3-x)^{2} = (x-3)^{2}. \] Substitute this back into the equation: \[ (x-3)^{2} \cdot x - 3(x-3)^{2}. \] ### Step 3: Factor Out \((x-3)^{2}\) Now, factor \((x-3)^{2}\) out of both terms: \[ (x-3)^{2}(x - 3). \] ### Step 4: Combine the Factors Multiply the terms inside the parentheses: \[ (x-3)^{2} \cdot (x - 3) = (x-3)^{3}. \] ### Conclusion Thus, we have shown that: \[ x(x-3)^{2} - 3(3-x)^{2} = (x-3)^{3}. \] **Final Answer:** After factoring, both terms share (x – 3)². Thus, x(x−3)² − 3(3−x)² = (x−3)²(x − 3) = (x−3)³.