Question

Y=x^4-x^3-5x^2-x-6 Show steps

Answer 1

Answer:

$(x-3)({x}^{2}+1)(x+2)$

Step-by-step explanation:

1) Factor $x^4 - x^3 - 5x^2 - x - 6$ using Polynomial Division.

1 - Factor the following.

$x^4 - x^3 - 5x^2 - x - 6$

2 - First, find all factors of the constant term 6.

$1, 2, 3, 6$

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..

$1^4 - 1^3 - 5 * 1^2 - 1 - 6 = - 12$

Substitute -1 into x. Since the result is not 0, x+1 is not a factor..

$( - 1 ) ^ 4 - ( - 1 ) ^ 3 - 5 ( - 1 ) ^ 2 + 1 - 6 = - 8$

Substitute 2 into x. Since the result is not 0, x-2 is not a factor..

${2}^{4}-{2}^{3}-5\times {2}^{2}-2-6 = -20$

Substitute -2 into x. Since the result is 0, x+2 is a factor..

${(-2)}^{4}-{(-2)}^{3}-5{(-2)}^{2}+2-6 = 0$

⇒ $x+2$

4 - Polynomial Division: Divide ${x}^{4}-{x}^{3}-5{x}^{2}-x-6$ by $x+2$

                                        $x^3$                $-3x^2$              $x$           $-3$

                                            --------------------------------------------------------------

$x+2$                                    |   $x^4$        $-x^3$               $-5x^2$     $-x$           $-6$

                                                  $x^4$         $2x^3$

                                                 ------------------------------------------------------------

                                               $-3x^3$         $-5x^2$          $-x$              $-6$

                                                 $-3x^3$       $-6x^2$

                                                   ---------------------------------------

                                                    $-3x$         $-6$

                                                       $-3x$      $-6$

                                                       ---------------

5 - Rewrite the expression using the above.

${x}^{3}-3{x}^{2}+x-3$

2) Factor out common terms in the first two terms, then in the last two terms.

$({x}^{2}(x-3)+(x-3))(x+2)$

3) Factor out the common term $x-3$.

$(x-3)({x}^{2}+1)(x+2)$

Answer 2

Answer:

The roots (zeros) are the x values where the graph intersects the x-axis. To find the roots (zeros), replace y with 0 and solve for x. x= i, −i, 3,−2

Step-by-step explanation: