Question

find all zeroes of the polynomial 4xpower 4 - 20 xcube + 23 x square + 5x-6 if two zeroes are 2 and 3

Answer 1

Here is your solution.

Given,

Quartic Polynomial = 4x⁴ - 20x³ + 23x² + 5x - 6.

Zeroes = 2 and 3.

We have to find 2 another zeroes.

Let's do it !

Let another 2 zeroes are α and ß.

Here,

Coefficient of x⁴ ( a ) = 4

Coefficient of x³ ( b ) = -20

Coefficient of x² ( c ) = 23

Coefficient of x ( d ) = 5

Constant term ( e ) = -6

We know the relationship between zeroes and coefficient of x of a Quatric Polynomial.

⇒ Sum of zeroes = -b / a

⇒ 2 + 3 + α + ß = - ( -20 ) / 4

⇒ 5 + α + ß = 20 / 4

⇒ 5 + α + ß = 5

⇒ α + ß = 5 - 5

∴ α + ß = 0   ----------- ( 1 )

Now,

⇒ Product of zeroes = e/a

⇒ 2 × 3 × α × ß = -6 / 4

⇒ 6αß = -6/4

⇒ αß = -6 ÷ ( 4 × 6 )

⇒ αß = -1/4.   --------- ( 2 )

Now,

⇒ ( α - ß )² = ( α + ß )² - 4αß

By substituting the values of ( α + ß ) and ( αß ).

⇒ ( α - ß )² = ( 0 )² - 4 ( -1/4 )

⇒ ( α - ß )² = 0 + 1

⇒ ( α - ß )² = 1

⇒ ( α - ß ) = √1

∴  ( α - ß ) = 1   ----------- ( 3 )

By adding ( 1 ) and ( 3 ) ,

⇒ α + ß + α - ß = 0 + 1

⇒ 2α = 1

∴  α = 1/2

Now,by substituting the value of α in ( 1 ),

⇒ α + ß = 0

⇒ ( 1/2 ) + ß = 0

∴  ß = -1/2

Hence,the another two zeroes are ( 1/2 ) and ( -1/2 ).


Hope it helps !!