Question

If two zeroes of the polynomial
${x}^{4} - 6 {x}^{3} - 26 {x}^{2} + 138x - 35$
are
$2 + - \sqrt{3}$
find the other zeroes.

Plz give in full details
​

Answer 1

Given :-

• Two zeros of the Polynomial x⁴ - 6x³ - 26x² + 138x - 35 = 0 are (2 ± √3).

To Find :-

• Other Zeros of Given Polynomial. ?

Concept used :-

• if a & b are the zeros of Polynomial than , (x - a)(x - b) is completely divide the given Polynomial .
• The Quadratic Equation Formed by sum & product of zeros is :- x² - (sum of Zeros)x + Product of zeros = 0

Solution :-

→ sum of Roots = (2 + √3) + (2 - √3) = 4

→ Product of Roots = (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1

Hence,

→ Quadratic Equation is = x² - 4x + 1 = 0

_______________

Now,

Divide :-

x² - 4x + 1 ) x⁴ - 6x³ - 26x² + 138x - 35 (x² - 2x - 35

x⁴ - 4x³ + x²_______

-2x³ - 27x² + 138x

-2x³ + 8x² - 2x

-35x² + 140x - 35

-35x² + 140x - 35

______0______

Now, Splitting The Quotient we get,

→ x² - 2x - 35 = 0

→ x² - 7x + 5x - 35 = 0

→ x(x - 7) + 5(x - 7) = 0

→ (x - 7)(x + 5) = 0

→ x = 7 & (-5). (Ans.)

Hence, Other zeros of given Polynomial are 7 & (-5).

Answer 2

Correct Question :

If two zeros of the polynomial 'x⁴ - 6x³ - 26x² + 138x - 35' are 2 ± √3 .Find the other zeros.

Solution :

According to remainder theorem, if two zeros of a polynomial are given then, (x - a)(x - b) will completely divide the polynomial and the remainder will be = 0 & quotient = 0

Now, finding divisor :

➝ [x + (- 2 - √3)][(x + (- 2 + √3)] = Divisor

➝ x² + (- 2 - √3 - 2 + √3)x + (-2 - √3) × (-2 + √3) = Divisor

➝ x² + (-4)x + (-2)² - (√3)² = Divisor

➝ x² - 4x + 4 - 3 = Divisor

➝ x² - 4x + 1 = Divisor

Now dividing the polynomial by (x² - 4x + 1)

x² - 4x + 1)x⁴ - 6x³ - 26x² + 138x - 35(x²-2x - 35

x⁴ - 4x³ + x²

(-) (+) (-)

-2x³ - 27x² + 138x - 35

-2x³ + 8x² - 2x

(+) (-) (+)

-35x² + 140x - 35

-35x² + 140x - 35

(+) (-) (-)

0

So, divisor = x² - 2x - 35 .

Therefore, x² - 2x - 35 = 0

➻ x² - 7x + 5x - 35 = 0

➻ x(x - 7) + 5(x - 7) = 0

➻ (x + 5)(x - 7) = 0

➻ x = -5 or, x = 7

➥ Other two zeros = -5 & 7

Therefore,

All the zeros of polynomial are (2 + √3),(2 - √3), -5 & 7 .