Factorise 1-16x^2+64x^4
Answers
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x4" was replaced by "x^4". 1 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
(1 - (16 • (x2))) + 26x4
STEP
2
:
Equation at the end of step
2
:
(1 - 24x2) + 26x4
STEP
3
:
Trying to factor by splitting the middle term
3.1 Factoring 64x4-16x2+1
The first term is, 64x4 its coefficient is 64 .
The middle term is, -16x2 its coefficient is -16 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 64 • 1 = 64
Step-2 : Find two factors of 64 whose sum equals the coefficient of the middle term, which is -16 .
-64 + -1 = -65
-32 + -2 = -34
-16 + -4 = -20
-8 + -8 = -16 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and -8
64x4 - 8x2 - 8x2 - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
8x2 • (8x2-1)
Add up the last 2 terms, pulling out common factors :
1 • (8x2-1)
Step-5 : Add up the four terms of step 4 :
(8x2-1) • (8x2-1)
Which is the desired factorization
Trying to factor as a Difference of Squares:
3.2 Factoring: 8x2-1
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 : 8 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Trying to factor as a Difference of Squares:
3.3 Factoring: 8x2-1
Check : 8 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Multiplying Exponential Expressions:
3.4 Multiply (8x2-1) by (8x2-1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (8x2-1) and the exponents are :
1 , as (8x2-1) is the same number as (8x2-1)1
and 1 , as (8x2-1) is the same number as (8x2-1)1
The product is therefore, (8x2-1)(1+1) = (8x2-1)2
Final result :
(8x2 - 1)2
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