Question

hola mates...

plz do in a notebook...

determine which of the following polynomials has (x+1) a factor :

1) x³+x²+x+1

2) x^4+3x³+3x²+x+1

3) x³-x²-(2+underroot 2)x+underroot 2

4) x^4x³+x²+x+1

only meaningful answers plz...

thanks for the help...

#Kratika

Answer 1

If ( x + 1 ) will be a factor, reminder will be 0.


By corollary 1 of the remainder theorem, x + 1 = { x - ( -1 ) } ,




1 ) : ( - 1 )³ + ( - 1 )² + ( - 1 ) + 1 = Remainder

- 1 + 1 - 1 + 1 = Remainder

0 = Remainder


remainder is 0, Therefore ( x + 1 ) is the factor of x³ + x² + x + 1



2 ):

( - 1 )⁴ + 3( - 1 )³ + 3( - 1 )² + ( - 1 ) + 1 = Remainder

1 - 3 + 3 - 1 + 1 = Remainder

1 = Remainder



remainder is 1 therefore ( x + 1 ) is not a factor of x^4+3x³+3x²+x+1



3 ):

( - 1 )³ - ( - 1 )² - ( 2 + √2 )( - 1 ) + √2 = Remainder

- 1 - 1 - 2 - √2 + √2 = Remainder

- 4 = Remainder



remainder is - 4 therefore ( x + 1 ) is not a factor of x³-x²-(2+underroot 2)x+underroot 2




4 ):

( - 1 )⁴ + ( - 1 )³ + ( - 1 )² + ( - 1 ) + 1 = Remainder

1 - 1 + 1 - 1 + 1 = Remainder

1 = Remainder


Remainder is 1 therefore ( x + 1 ) is not a factor of x⁴+ x³+x²+x+1

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Answer 2

Hey mate!!!

your answer is attached above !!


Doubts are surely welcome !!!