Question

what is the factor of x^3-7x^2+11x-5

Answer 1

x³ - 7x² + 11x - 5


Splitting - 7x² in two terms.

- 7x² = - x² - 6x²


x³ - 7x² + 11x - 5

x³ - x² - 6x² + 11x - 5


$\text{If we notice, we can take} \: x^{2} \text{ common in} \: ( x^{ 3} - x^{2} ) \\ \text{and we can write} - 6x^{2} + 11x - 5 \: \: \text{as} \: \: \bold{ - [ 6x^{2} - 11x + 5 ]}$

So,


x²( x - 1 ) - [ 6x² - 11x + 5 ]


Then,
6x² + 11x + 5 is an quadratic equation which can be solved by splitting middle term method. in the equation middle term is 11x, according to the question 11x on splitting, will be 6x + 5x.


x²( x - 1 ) - [ 6x² - 6x - 5x + 5 ]

x²( x - 1 ) - [ 6x( x - 1 ) + 5( x - 1 ) ]

x²( x - 1 ) - ( x - 1 )( 6x + 5 )

( x - 1 )( x² - 6x + 5 )

( x - 1 ) { x² - 5x - x + 5 }

( x - 1 ){ x( x - 5 ) - ( x - 5 ) }

( x - 1 )( x - 1 )( x - 5 )

( x - 5 )( x - 1 )²



Thus,
factors of x^3-7x^2+11x-5 are ( x - 5 ) and ( x - 1 )