If two zeroes of the polynomial …

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If two zeroes of the polynomial x⁴-6x³-26x²+138x-35 are 2+-√3. find other zeroes.

Posted by Hrithik Singh 6 years, 3 months ago

CBSE > Class 11 > Mathematics

2 answers

Sia ? 6 years, 3 months ago

Two zeros are {tex}2\pm\sqrt3{/tex}
Sum of Zeroes {tex}2 + \sqrt { 3 } + 2 - \sqrt { 3 } = 4{/tex}
and product of zeroes = {tex}( 2 + \sqrt { 3 } ) ( 2 - \sqrt { 3 } ) = 4 - 3 = 1{/tex}
Hence quadratic polynomial formed out of this will be a factor of given polynomial,
So, x² - (sum of zeroes)x + product of zeroes
= x² - 4x + 1 will be a factor of given polynomial,
Divide given polynomial with x² - 4x + 1 to get other zeroes.

Now,
x² -2x - 35
= x² - 7x + 5x - 35
= x(x - 7) + 5(x - 7)
= (x - 5) (x - 7)
{tex}\therefore {/tex} Zeros are
x = 7 and x = -5
{tex}\therefore {/tex} Other two zeros are 7 and -5

2Thank You

Shantanu Kapse 6 years, 3 months ago

Let the remaining two zeros be X and Y, As, Sum of roots =6 , product of roots = -35 We get 2 equations, 2+√3 + 2-√3 + x +y = 6 x+y= 2-----(1) (2+√3)(2-√3)(xy) = -35 xy=-35 x=-35/y -----(2) Subs (2) in (1) y-35/y =2 y²-2y-35=0 y=[2±√(144)]/2 y=7 y=-5 x=-5 x=7 Hence the roots are (2±√3),(-5),(7) .

0Thank You

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