Question

(x^4-6x^2-27)/(x+2)
Plz help im a returd

Answer 1

Answer:(x2 + 3) • (x + 3) • (x - 3)

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            x + 2      

Step-by-step explanation:

Step  1  :

Equation at the end of step  1  :

 

Step  2  :

           x4 - 6x2 - 27

Simplify   —————————————

               x + 2    

Trying to factor by splitting the middle term

2.1     Factoring  x4 - 6x2 - 27  

The first term is,  x4  its coefficient is  1 .

The middle term is,  -6x2  its coefficient is  -6 .

The last term, "the constant", is  -27  

Step-1 : Multiply the coefficient of the first term by the constant   1 • -27 = -27  

Step-2 : Find two factors of  -27  whose sum equals the coefficient of the middle term, which is   -6 .

     -27    +    1    =    -26  

     -9    +    3    =    -6    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -9  and  3  

                    x4 - 9x2 + 3x2 - 27

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x2 • (x2-9)

             Add up the last 2 terms, pulling out common factors :

                   3 • (x2-9)

Step-5 : Add up the four terms of step 4 :

                   (x2+3)  •  (x2-9)

            Which is the desired factorization

Polynomial Roots Calculator :

2.2    Find roots (zeroes) of :       F(x) = x2+3

Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  3.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,3

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        4.00      

     -3       1        -3.00        12.00      

     1       1        1.00        4.00      

     3       1        3.00        12.00      

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

2.3      Factoring:  x2-9  

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

        A2 - AB + BA - B2 =

        A2 - AB + AB - B2 =

        A2 - B2

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 9 is the square of 3

Check :  x2  is the square of  x1  

Factorization is :       (x + 3)  •  (x - 3)  

Final result :

 (x2 + 3) • (x + 3) • (x - 3)

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            x + 2            

Processing ends successfully