Answer:
[tex](x-3)({x}^{2}+1)(x+2)[/tex]
Step-by-step explanation:
1) Factor [tex]x^4 - x^3 - 5x^2 - x - 6[/tex] using Polynomial Division.
1 - Factor the following.
[tex]x^4 - x^3 - 5x^2 - x - 6[/tex]
2 - First, find all factors of the constant term 6.
[tex]1, 2, 3, 6[/tex]
3 - Try each factor above using the Remainder Theorem.
Substitute 1 into x. Since the result is not 0, x-1 is not a factor..
[tex]1^4 - 1^3 - 5 * 1^2 - 1 - 6 = - 12[/tex]
Substitute -1 into x. Since the result is not 0, x+1 is not a factor..
[tex]( - 1 ) ^ 4 - ( - 1 ) ^ 3 - 5 ( - 1 ) ^ 2 + 1 - 6 = - 8[/tex]
Substitute 2 into x. Since the result is not 0, x-2 is not a factor..
[tex]{2}^{4}-{2}^{3}-5\times {2}^{2}-2-6 = -20[/tex]
Substitute -2 into x. Since the result is 0, x+2 is a factor..
[tex]{(-2)}^{4}-{(-2)}^{3}-5{(-2)}^{2}+2-6 = 0[/tex]
⇒ [tex]x+2[/tex]
4 - Polynomial Division: Divide [tex]{x}^{4}-{x}^{3}-5{x}^{2}-x-6[/tex] by [tex]x+2[/tex]
[tex]x^3[/tex] [tex]-3x^2[/tex] [tex]x[/tex] [tex]-3[/tex]
--------------------------------------------------------------
[tex]x+2[/tex] | [tex]x^4[/tex] [tex]-x^3[/tex] [tex]-5x^2[/tex] [tex]-x[/tex] [tex]-6[/tex]
[tex]x^4[/tex] [tex]2x^3[/tex]
------------------------------------------------------------
[tex]-3x^3[/tex] [tex]-5x^2[/tex] [tex]-x[/tex] [tex]-6[/tex]
[tex]-3x^3[/tex] [tex]-6x^2[/tex]
---------------------------------------
[tex]-3x[/tex] [tex]-6[/tex]
[tex]-3x[/tex] [tex]-6[/tex]
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5 - Rewrite the expression using the above.
[tex]{x}^{3}-3{x}^{2}+x-3[/tex]
2) Factor out common terms in the first two terms, then in the last two terms.
[tex]({x}^{2}(x-3)+(x-3))(x+2)[/tex]
3) Factor out the common term [tex]x-3[/tex].
[tex](x-3)({x}^{2}+1)(x+2)[/tex]
