Question

Obtain other zeroes of the polynomial p(x) = 2x+-x-11x2 + 5x + 5 if two of its zeores are root+ 5 and root-5

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Answer 1

CORRECT QUESTION.

Obtain other zeroes of the polynomial

p(x) = 2x⁴ - x³ - 11x² + 5x + 5 if thr two zeroes

are = √5 and - √5

EXPLANATION.

zeroes of the polynomial = √5 and -√5

=> x = √5 and x = -√5

=> products of the zeroes of the polynomial

=> ( x - √5 ) ( x + √5 )

=> x² - 5

using identities = ( a + b) ( a - b) = a² - b²

divide the polynomial = 2x⁴ - x³ - 11x² + 5x + 5

by = x² - 5

we get,

=> 2x² - x - 1

by factorise the polynomial into middle

term split

we get,

=> 2x² - x - 1 = 0

=> 2x² - 2x + x - 1 = 0

=> 2x ( x - 1 ) + 1 ( x - 1 ) = 0

=> ( 2x + 1 ) ( x - 1 ) = 0

=> x = -1/2 and x = 1

Therefore,

All zeroes are = √5 , -√5 , -1/2 , 1

Answer 2

Step-by-step explanation:

Zeros are (x + √5)(x - √5) = x² - 5

Used identity: (a - b)(a + b) = a² - b²

x²-5 ) 2x⁴ - x³ - 11x² + 5x +5 ( 2x²-x-1

...........2x⁴ ....... - 10x²

____________________

.............. - x³ - x² + 5x + 5

............. - x³ ... ... + 5x

________________

................... - x² + 5

................... - x² + 5

______________

.................... ... 0

Now,

→ 2x² - x - 1 = 0

Split the middle term

→ 2x² - 2x + x - 1 = 0

→ 2x(x + 1) +1(x - 1) = 0

→ (2x + 1)(x - 1) = 0

On comparing we get,

→ x = -1/2, 1

Hence, the zeros are +√5, -√5, -1/2 and 1.