Math, asked by sweetie59, 1 year ago

x^4-1; x^3-11x^2+x-11​

Answers

Answered by joybiswas100
2

Step-by-step explanation:

Final result :

(x2 + 1) • (x + 3) • (x - 4)

Step by step solution :

Step 1 :

Equation at the end of step 1 :

((((x4)-(x3))-11x2)-x)-12

Answered by hannah100
4

:Final result :

 (x2 + 1) • (x + 3) • (x - 4)

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 ((((x4)-(x3))-11x2)-x)-12

Step  2  :

Polynomial Roots Calculator :

2.1    Find roots (zeroes) of :       F(x) = x4-x3-11x2-x-12

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  -12.  

The factor(s) are:  

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,12  

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        -20.00      

     -2       1        -2.00        -30.00      

     -3       1        -3.00        0.00      x+3  

     -4       1        -4.00        136.00      

     -6       1        -6.00        1110.00      

     -12       1       -12.00       20880.00      

     1       1        1.00        -24.00      

     2       1        2.00        -50.00      

     3       1        3.00        -60.00      

     4       1        4.00        0.00      x-4  

     6       1        6.00        666.00      

     12       1        12.00       17400.00      

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms  

In our case this means that  

  x4-x3-11x2-x-12  

can be divided by 2 different polynomials,including by  x-4  

Polynomial Long Division :

2.2    Polynomial Long Division  

Dividing :  x4-x3-11x2-x-12  

                             ("Dividend")

By         :    x-4    ("Divisor")

dividend     x4  -  x3  -  11x2  -  x  -  12  

- divisor  * x3     x4  -  4x3              

remainder         3x3  -  11x2  -  x  -  12  

- divisor  * 3x2         3x3  -  12x2          

remainder             x2  -  x  -  12  

- divisor  * x1             x2  -  4x      

remainder                 3x  -  12  

- divisor  * 3x0                 3x  -  12  

remainder                    0

Quotient :  x3+3x2+x+3  Remainder:  0  

Polynomial Roots Calculator :

2.3    Find roots (zeroes) of :       F(x) = x3+3x2+x+3

    See theory in step 2.1  

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        0.00      x+3  

     1       1        1.00        8.00      

     3       1        3.00        60.00      

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms  

In our case this means that  

  x3+3x2+x+3  

can be divided with  x+3  

Polynomial Long Division :

2.4    Polynomial Long Division  

Dividing :  x3+3x2+x+3  

                             ("Dividend")

By         :    x+3    ("Divisor")

dividend     x3  +  3x2  +  x  +  3  

- divisor  * x2     x3  +  3x2          

remainder             x  +  3  

- divisor  * 0x1                  

remainder             x  +  3  

- divisor  * x0             x  +  3  

remainder                0

Quotient :  x2+1  Remainder:  0  

Polynomial Roots Calculator :

2.5    Find roots (zeroes) of :       F(x) = x2+1

    See theory in step 2.1  

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

The factor(s) are:  

of the Leading Coefficient :  1

of the Trailing Constant :  1  

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        2.00      

     1       1        1.00        2.00      

Polynomial Roots Calculator found no rational roots

Final result :

 (x2 + 1) • (x + 3) • (x - 4)

Processing ends successfully

Step-by-step explanation:

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