1. **Problem Statement:** Find the H.C.F. (Highest Common Factor) and L.C.M. (Least Common Multiple) of the polynomials: $$x^3(x + 7) \quad \text{and} \quad x^4(x - 7)^2$$ 2. **Formula and Rules:** - The H.C.F. of two polynomials is the product of the lowest powers of all common factors. - The L.C.M. of two polynomials is the product of the highest powers of all factors appearing in either polynomial. 3. **Step-by-step Solution:** - First polynomial: $$x^3(x + 7) = x^3(x + 7)$$ - Second polynomial: $$x^4(x - 7)^2 = x^4(x - 7)^2$$ - Identify common factors: - Both have powers of $$x$$: $$x^3$$ and $$x^4$$ - The binomials $$x + 7$$ and $$x - 7$$ are different, so no common binomial factor. - H.C.F. is the product of the lowest powers of common factors: - For $$x$$, lowest power is $$x^3$$ - No common binomial factor - So, $$\text{H.C.F.} = x^3$$ - L.C.M. is the product of the highest powers of all factors: - For $$x$$, highest power is $$x^4$$ - Include both binomials $$x + 7$$ and $$x - 7$$ with their powers: - $$x + 7$$ to power 1 - $$x - 7$$ to power 2 - So, $$\text{L.C.M.} = x^4 (x + 7)(x - 7)^2$$ 4. **Final Answer:** $$\boxed{\text{H.C.F.} = x^3}$$ $$\boxed{\text{L.C.M.} = x^4 (x + 7)(x - 7)^2}$$