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InterviewSolution
| 1. |
If the zeros of the polynomial f (x)=2x-15x+3x-30 are in A.P., find them |
| Answer» Let, {tex}α{/tex} = a - d, {tex}β{/tex} = a and {tex}\\gamma {/tex} = a + d be the zeroes of the polynomial.f(x) = 2x3 - 15x2 + 37x - 30{tex}\\alpha + \\beta + \\gamma = - \\left( {\\frac{{ - 15}}{2}} \\right) = \\frac{{15}}{2}{/tex} ....... (i){tex}\\alpha \\beta \\gamma = - \\left( {\\frac{{ - 30}}{2}} \\right) = 15{/tex} ......... (ii)From (i)a - d + a + a + d = {tex}\\frac{{15}}{2}{/tex}\xa0So, 3a = {tex}\\frac{{15}}{2}{/tex}a = {tex}\\frac{{5}}{2}{/tex}and From (ii)a(a - d)(a + d) = 15So, a(a2 - d2) = 15{tex}⇒ \\frac{{5}}{2}{/tex}{tex}\\left[\\left(\\frac52\\right)^2\\;-d^2\\right]{/tex} = 15{tex}⇒ \\frac{25}4\\;-d^2{/tex}= 6{tex}⇒ \\;d^2\\;=\\;\\frac{25}4-6\\;{/tex}{tex}⇒ {d^2} = \\frac{1}{4}{/tex}{tex}⇒ d = \\frac{1}{2}{/tex}Therefore, {tex}\\alpha = \\frac{5}{2} - \\frac{1}{2} = \\frac{4}{2} = 2{/tex}{tex}\\beta = \\frac{5}{2}{/tex}{tex}\\gamma = \\frac{5}{2} + \\frac{1}{2} = 3{/tex}. | |