Question

If 1 and 2 are zeros of polynomials of x^4-4x^3-15x^2+58x-40 find other zeros if any

Answer 1

Hey Friend ☺

x^4 - 4x^3 - 15x^2 + 58x - 40 = 0

》x^4 - 5x^3 + x^3 - 15x^2 + 58x - 40

》x^3 ( x - 5 ) + x^3 - 15x^2 + 58x - 40

》x^3 ( x - 5 ) + x^3 - 5x^2 - 10x^2 + 58x - 40

》x^3 ( x - 5 ) + x^2 ( x - 5 ) - 10x^2 + 58x - 40

》x^3 ( x - 5 ) + x^2 ( x- 5 ) - 10x^2 + 50x + 8x - 40

》x^3 ( x - 5 ) + x^2 ( x - 5 ) - 10x ( x - 5 ) + 8 ( x - 5 )

Taking ( x - 5 ) as common we get

》( x - 5 ) ( x^3 + x^2 - 10x + 8 )

》( x - 5 ) ( x^3 - 2x^2 + 3x^2 - 10x + 8 )

》( x - 5 ) [ x^2 ( x - 2 ) + 3x^2 - 10x + 8 ]

》( x - 5 ) [ x^2 ( x - 2 ) + 3x^2 - 6x - 4x + 8]

》( x - 5 ) [ x^2 ( x - 2 ) + 3x ( x - 2 ) - 4 ( x - 2 ) ]

》( x - 5 )( x - 2 )( x^2 + 3x - 4 )

》( x - 5 )( x - 2 )( x^2 + 4x - x - 4 )

》( x - 5 )( x - 2 )[ x ( x + 4 ) + 1 ( x + 4 ) ]

》( x - 5 )( x - 2 )( x + 4 )( x + 1 ) = 0

So other zeroes of the equation are 5 & - 4

Hope it helps you ..!!

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