JEEAdvanced
Real numbers a, b, c satisfy a + b + c = 6, a^2 + b^2 + c^2 = 14, and…
Question
Real numbers a, b, c satisfy a + b + c = 6, a^2 + b^2 + c^2 = 14, and a^3 + b^3 + c^3 = 36. Find a^4 + b^4 + c^4.
✓ Verified answer: 98checked by our engine — not a guess
Step-by-step solution
Let e1, e2, e3 be the elementary symmetric polynomials and p_k the power sums.
e1 = 6. From p2 = e1^2 - 2e2: 14 = 36 - 2e2, so e2 = 11.
Newton: p3 = e1 p2 - e2 p1 + 3 e3 ⇒ 36 = 6·14 - 11·6 + 3e3 = 84 - 66 + 3e3 = 18 + 3e3, so e3 = 6.
Then p4 = e1 p3 - e2 p2 + e3 p1 = 6·36 - 11·14 + 6·6 = 216 - 154 + 36 = 98.
(Indeed a,b,c are 1,2,3, and 1+16+81 = 98.)
Final answer98
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Algebra solutions
Let a, b, c be the (possibly complex) roots of the cubic x^3 - 5x^2 +…JEE · MathA function f defined for all nonzero real x satisfies f(x) + 2 f(1/x)…JEE · MathLet a, b, c be positive real numbers with abc = 8. Find the minimum…JEE · MathA polynomial P(x) leaves remainder 3 on division by (x - 1),…JEE · MathThe cubic x^3 - a x^2 + (a - 1)x - 5 = 0 (with a a real parameter)…JEE · MathLet w be the real number w = cbrt(20 + 14√2) + cbrt(20 - 14√2), where…JEE · MathEvaluate the sum S = 1·1! + 2·2! + 3·3! + 4·4! + 5·5! + 6·6!, where…JEE · MathFind the greatest integer not exceeding (2 + √5)^5; that is, compute…JEE · Math