Question

find all the zeroes of 2xfour power-3xcube-3xsquare +6x-2if you know that two of its zeroes are root 2 and -root 2​

Answer 1

Solution :-

Given :-

√2 and - √2 are the 2 zeroes of 2x^4 - 3x³ - 3x² + 6x - 2

We can find other 2 zeroes If we divide 2x^4 - 3x³ - 3x² + 6x - 2 by polynomial whose zeroes are √2 and - √2

Finding the polynomial whose zeroes are √2 and - √2

Zeroes of a polynomial √2 , - √2

So, let α = √2 and β = - √2

Quadratic polynomial = (x - α)(x - β)

= (x - √2){ x - ( - √2) }

= (x - √2)(x + √2)

Using (x + y)(x - y) = x² - y² identity

= x² - (√2)²

= x² - 2

Therefore the polynomial whose zeroes are √2 and -√2 is x² - 2

Now, divide 2x^4 - 3x³ - 3x² + 6x - 2

[ Refer to attachment ]

So, by division algorithm

2x^4 - 3x³ - 3x² + 6x - 2 = (x² - 2)(2x² - 3x + 1)

Therefore we can get other 2 zeroes by factorising 2x² - 3x + 1

2x² - 3x + 1

Splitting the middle term

= 2x² - 2x - x + 1

= 2x(x - 1) - 1(x - 1)

= (2x - 1)(x - 1)

Now equate it to 0 to find the zeroes

⇒ (2x - 1)(x - 1) = 0

⇒ 2x - 1 = 0 or x - 1 = 0

⇒ 2x = 1 or x = 1

⇒ x = 1/2 or x = 1

Hence, all the zeroes of the given polyinomial are √2, - √2, 1/2, 1.