How do you factor [tex]x^4 - x^3 - 7x^2 - x + 6[/tex]?

x 4 − x 3 − 7 x 2 − x + 6 = x 4 − x 3 − 6 x 2 − x 2 − x + 6 = = x 2 ( x 2 − x − 6 ) − ( x 2 − x − 6 ) = ( x 2 − 1 ) ( x 2 − x − 6 ) = = ( x − 1 ) ( x + 1 ) ( x 2 − 3 x + 2 x − 6 ) = = ( x − 1 ) ( x + 1 ) [ x ( x − 3 ) + 2 ( x − 3 )] = ( x − 1 ) ( x + 1 ) ( x − 3 ) ( x + 2 )

Answered by kate200468 | 2024-06-10

To factor the polynomial x 4 − x 3 − 7 x 2 − x + 6 , we first find rational roots using the Rational Root Theorem, discovering that x − 3 is a root. After synthetic division, we arrive at the factors ( x − 3 ) ( x + 2 ) ( x − 1 ) ( x + 1 ) . This gives us the fully factored form of the polynomial.
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Answered by kate200468 | 2024-12-18