Factorise X6-25x4 +16x2-400
Answers
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: