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x 3 − x 2 − 4 x + 4 = x 2 ( x − 1 ) − 4 ( x − 1 ) = ( x 2 − 4 ) ( x − 1 ) = ( x − 2 ) ( x + 2 ) ( x − 1 )

Answered by konrad509 | 2024-06-10

The fully factored form of the polynomial x 3 − x 2 − 4 x + 4 over the set of complex numbers is:
x 3 − x 2 − 4 x + 4 = ( x − 1 ) ( x + 2 ) ( x − 2 ) .
To factor the polynomial x 3 − x 2 − 4 x + 4 completely over the set of complex numbers , we can use either long division or synthetic division to find one of its roots (zeros).
Once we find one root, we can factor the polynomial using the factor theorem o**r synthetic division **repeatedly to find all its roots.
We have to find one** root** first by trying some values for x.
We know that if a value of x makes the polynomial equal to zero, it is a root (zero) of the polynomial.
By trying different values of x, we find that x = 1 is a root of the polynomial:
( 1 ) 3 − ( 1 ) 2 − 4 ( 1 ) + 4 = 1 − 1 − 4 + 4 = 0
Now that we have one root, x = 1, we can factor the polynomial using synthetic division :
( x 3 − x 2 − 4 x + 4 ) / ( x − 1 )
The quotient is x 2 + 0 x − 4 , which simplifies to x 2 − 4 .
Now, we can further factor x 2 − 4 using the difference of squares:
x 2 − 4 = ( x + 2 ) ( x − 2 )
So, the fully factored form of the polynomial x 3 − x 2 − 4 x + 4 over the set of complex numbers is:
x 3 − x 2 − 4 x + 4 = ( x − 1 ) ( x + 2 ) ( x − 2 ) .
Know more about** Polynomials **here:
brainly.com/question/2833285
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Answered by duttakoushik094 | 2024-06-17

The polynomial x 3 − x 2 − 4 x + 4 can be completely factored over the complex numbers as ( x − 1 ) ( x − 2 ) ( x + 2 ) . The roots of the polynomial are 1, 2, and -2. This can be achieved by finding a root, using synthetic division to reduce the polynomial, and then factoring further.
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Answered by duttakoushik094 | 2024-10-01

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Answered by ayuliati054 | 2025-08-22