\[ b^3 + c^3 = (b + c)(b^2 - bc + c^2) = 12 \cdot 39 = 468 \]

["Understanding the Identity: ( b^3 + c^3 = (b + c)(b^2 - bc + c^2) = 468 ) with ( b + c = 12 ) and ( b^3 + c^3 = 468 )", "The algebraic identity\n[\nb^3 + c^3 = (b + c)(b^2 - bc + c^2)\n]\nis a cornerstone in simplifying cubic expressions and solving equations involving sums and powers of variables. This article explores a specific application of this identity, where ( b + c = 12 ) and ( b^3 + c^3 = 468 ), and demonstrates how to find possible integer or rational values of ( b ) and ( c ) that satisfy the equation.", "---", "### What is the Identity Behind ( b^3 + c^3 = (b + c)(b^2 - bc + c^2) )?", "This identity stems from expanding ( (b + c)^3 ) and isolating the cubic sum:\n[\n(b + c)^3 = b^3 + c^3 + 3bc(b + c)\n]\nRearranging gives:\n[\nb^3 + c^3 = (b + c)^3 - 3bc(b + c)\n]\nFactoring out ( (b + c) ), we get:\n[\nb^3 + c^3 = (b + c)\left( (b + c)^2 - 3bc \right) = (b + c)(b^2 - bc + c^2)\n]\nThis formula is particularly useful for factoring sums of cubes and solving equations involving symmetric expressions in ( b ) and ( c ).", "---", "### Given Values", "We are given:\n[\nb + c = 12 \quad \ ext{and} \quad b^3 + c^3 = 468\n]\nUsing the identity:\n[\n468 = (12)(b^2 - bc + c^2)\n]\nDivide both sides by 12:\n[\nb^2 - bc + c^2 = \frac{468}{12} = 39\n]", "---", "### Express ( b^2 - bc + c^2 ) in Terms of ( b + c ) and ( bc )", "Recall the identity:\n[\n(b + c)^2 = b^2 + 2bc + c^2\n]\nSo,\n[\nb^2 + c^2 = (b + c)^2 - 2bc = 12^2 - 2bc = 144 - 2bc\n]\nSubstitute into the expression ( b^2 - bc + c^2 ):\n[\nb^2 - bc + c^2 = (b^2 + c^2) - bc = (144 - 2bc) - bc = 144 - 3bc\n]", "Set equal to 39:\n[\n144 - 3bc = 39\n]\nSolve for ( bc ):\n[\n3bc = 144 - 39 = 105 \quad \Rightarrow \quad bc = 35\n]", "---", "### Solving for ( b ) and ( c )", "Now we know:\n[\nb + c = 12, \quad bc = 35\n]\nThese are the sum and product of two numbers, so ( b ) and ( c ) are the roots of the quadratic equation:\n[\nx^2 - (b + c)x + bc = 0 \quad \Rightarrow \quad x^2 - 12x + 35 = 0\n]", "Solve using the quadratic formula:\n[\nx = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 35}}{2} = \frac{12 \pm \sqrt{144 - 140}}{2} = \frac{12 \pm \sqrt{4}}{2} = \frac{12 \pm 2}{2}\n]\nSo,\n[\nx = \frac{14}{2} = 7 \quad \ ext{or} \quad x = \frac{10}{2} = 5\n]", "---", "### Solution: Possible Pairs ( (b, c) )", "The solutions are:\n[\n(b, c) = (7, 5) \quad \ ext{or} \quad (5, 7)\n]", "Check the result:\n[\nb^3 + c^3 = 7^3 + 5^3 = 343 + 125 = 468 \quad \ ext{✓}\n]\n[\nb + c = 7 + 5 = 12 \quad \ ext{✓}\n]", "---", "### Why This Identity is Useful in Algebra and Problem Solving", "- Factoring complex expressions: Breaking ( b^3 + c^3 ) into ( (b + c)(\cdots) ) simplifies computation.\n- Solving symmetric equations: When both sum and sum of cubes are given, expressing ( b^3 + c^3 ) in terms of ( b + c ) and ( bc ) enables solving for roots.\n- Applications in geometry and number theory: The identity helps factor sums of cubes in integer solutions or Pythagorean-like problems.", "---", "### Conclusion", "The identity ( b^3 + c^3 = (b + c)(b^2 - bc + c^2) ) is more than a formula—it’s a powerful tool for solving algebraic equations involving cubic terms. By combining the known sum ( b + c = 12 ) and the given cubic sum ( b^3 + c^3 = 468 ), we systematically derived the product ( bc = 35 ) and solved the resulting quadratic equation. The solution yields integer pairs ( (b, c) = (7, 5) ) or ( (5, 7) ), confirming the identity’s validity in concrete numerical cases.", "Understanding such relationships strengthens problem-solving skills in algebra, especially when dealing with symmetric expressions and polynomial identities.", "---", "Keywords for SEO:\n( b^3 + c^3 = (b + c)(b^2 - bc + c^2) = 468 ), sum of cubes identity, solve ( b + c = 12 ), ( b^3 + c^3 = 468 ), quadratic equation from sum and product, algebraic identities, solving cubic expressions, algebraic problem solving.", "---", "Related Reads:\n- How to factor ( b^3 + c^3 ) using sum of cubes formula\n- Solving symmetric equations with ( b + c = S ) and ( b^3 + c^3 = K )\n- Applications of ( b^2 - bc + c^2 ) in number theory and geometry", "---", "Understanding algebraic identities transforms complex equations into solvable systems—master them to unlock faster, deeper math mastery."]