Math, asked by brianchongo, 11 months ago

Factorise X6-25x4 +16x2-400

Answers

Answered by madhunisha05
1

Answer:

1 result(s) found

(x  

2

+4)⋅(x+2)⋅(x−2)⋅(x+5)⋅(x−5)

See steps

Step by Step Solution:

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STEP

1

:

Equation at the end of step 1

 (((x6)-(25•(x4)))-24x2)+400

STEP  

2

:

Equation at the end of step

2

:

 (((x6) -  52x4) -  24x2) +  400

STEP

3

:

Checking for a perfect cube

3.1    x6-25x4-16x2+400  is not a perfect cube

Trying to factor by pulling out :

3.2      Factoring:  x6-25x4-16x2+400  

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -16x2+400  

Group 2:  x6-25x4  

Pull out from each group separately :

Group 1:   (x2-25) • (-16)

Group 2:   (x2-25) • (x4)

              -------------------

Add up the two groups :

              (x2-25)  •  (x4-16)  

Which is the desired factorization

Trying to factor as a Difference of Squares:

3.3      Factoring:  x4-16  

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 : 16 is the square of 4

Check :  x4  is the square of  x2  

Factorization is :       (x2 + 4)  •  (x2 - 4)  

Polynomial Roots Calculator :

3.4    Find roots (zeroes) of :       F(x) = x2 + 4

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  4.

The factor(s) are:

of the Leading Coefficient :  1

of the Trailing Constant :  1 ,2 ,4

Let us test ....

  P    Q    P/Q    F(P/Q)     Divisor

     -1       1        -1.00        5.00      

     -2       1        -2.00        8.00      

     -4       1        -4.00        20.00      

     1       1        1.00        5.00      

     2       1        2.00        8.00      

     4       1        4.00        20.00      

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares:

3.5      Factoring:  x2 - 4  

Check : 4 is the square of 2

Check :  x2  is the square of  x1  

Factorization is :       (x + 2)  •  (x - 2)  

Trying to factor as a Difference of Squares:

3.6      Factoring:  x2 - 25  

Check : 25 is the square of 5

Check :  x2  is the square of  x1  

Factorization is :       (x + 5)  •  (x - 5)  

Final result :

 (x2+4)•(x+2)•(x-2)•(x+5)•(x-5)

Step-by-step explanation:

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