find the value of l and m if 8x^3+lx^2-27x+m is divisible by 2x^2-x-6
Answers
Solution :-
since 8x^3+Lx^2-27x+M is divisible by 2x^2-x-6 , it will also divisible by factors of 2x² - x - 6 .
so,
→ 2x² - x - 6 = 0
→ 2x² - 4x + 3x - 6 = 0
→ 2x(x - 2) + 3(x - 2) = 0
→ (x - 2)(2x + 3) = 0
→ x = 2 and (-3/2)
so, putting x = 2 we get,
→ p(2) = 8(2)³ + L(2)² - 27*2 + M
→ p(2) = 64 + 4L - 54 + M
→ p(2) = 4L + M + 10
since it is factor or p(x), remainder is equal to 0 .
→ 4L + M + 10 = 0
→ 4L + M = (-10) ------------- Eqn.(1)
similarly,
→ p(-3/2) = 8(-3/2)² + L(-3/2)² - 27(-3/2) + M
→ p(-3/2) = 18 + (9L/4) + (81/2) + M
again,
→ (72 + 9L + 162 + 4M) / 4 = 0
→ 9L + 4M = (-234) ---------- Eqn.(2)
multiply Eqn.(1) by 4 and subtracting from Eqn.(2),
→ (9L + 4M) - 4(4L + M) = (-234) - 4(-10)
→ 9L - 16L + 4M - 4M = (-234) + 40
→ (-7L) = (-194)
→ L = (194/7)
putting value of L in Eqn.(1) now,
→ 4(194/7) + M = (-10)
→ M = (-10) - (776/7)
→ M = (-70 - 776)/7
→ M = (-846/7)
Learn more :-
JEE mains Question :-
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. Find all the zeroes of the polynomial x4
– 5x3 + 2x2+10x-8, if two of its zeroes are 4 and 1.
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Answer:
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