f(x)=16(x⁴-64x) solution
Answers
Answer:
Trying to factor as a Difference of Cubes:
2.2 Factoring: x3 - 64
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 64 is the cube of 4
Check : x3 is the cube of x1
Factorization is :
(x - 4) • (x2 + 4x + 16)
Trying to factor by splitting the middle term
2.3 Factoring x2 + 4x + 16
The first term is, x2 its coefficient is 1 .
The middle term is, +4x its coefficient is 4 .
The last term, "the constant", is +16
Step-1 : Multiply the coefficient of the first term by the constant 1 • 16 = 16
Step-2 : Find two factors of 16 whose sum equals the coefficient of the middle term, which is 4 .
-16 + -1 = -17
-8 + -2 = -10
-4 + -4 = -8
-2 + -8 = -10
-1 + -16 = -17
1 + 16 = 17
2 + 8 = 10
4 + 4 = 8
8 + 2 = 10
16 + 1 = 17
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
2
:
x - 16x • (x - 4) • (x2 + 4x + 16) = 0
STEP
3
:
STEP
4
:
Pulling out like terms
4.1 Pull out like factors :
1025x - 16x4 = -x • (16x3 - 1025)
Trying to factor as a Difference of Cubes:
4.2 Factoring: 16x3 - 1025
Check : 16 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = 16x3 - 1025
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 16 and the Trailing Constant is -1025.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8 ,16
of the Trailing Constant : 1 ,5 ,25 ,41 ,205 ,1025