Reduce the rational ere pression
to Tonest form:
4²-4x+4
2 x²-8
Answers
Answer:
STEP
1
:
Equation at the end of step 1
((2 • (x4)) - 23x2) - 64
STEP
2
:
Equation at the end of step
2
:
(2x4 - 23x2) - 64
STEP
3
:
STEP
4
:
Pulling out like terms
4.1 Pull out like factors :
2x4 - 8x2 - 64 = 2 • (x4 - 4x2 - 32)
Trying to factor by splitting the middle term
4.2 Factoring x4 - 4x2 - 32
The first term is, x4 its coefficient is 1 .
The middle term is, -4x2 its coefficient is -4 .
The last term, "the constant", is -32
Step-1 : Multiply the coefficient of the first term by the constant 1 • -32 = -32
Step-2 : Find two factors of -32 whose sum equals the coefficient of the middle term, which is -4 .
-32 + 1 = -31
-16 + 2 = -14
-8 + 4 = -4 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and 4
x4 - 8x2 + 4x2 - 32
Step-4 : Add up the first 2 terms, pulling out like factors :
x2 • (x2-8)
Add up the last 2 terms, pulling out common factors :
4 • (x2-8)
Step-5 : Add up the four terms of step 4 :
(x2+4) • (x2-8)
Which is the desired factorization
Polynomial Roots Calculator :
4.3 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:
4.4 Factoring: x2-8
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.
Final result :
2 • (x2 + 4) • (x2 - 8)