divide (x^4 - 16) by x^3 + 2x^2 + 4x + 8
Answers
Answer:
x-2
Step-by-step explanation:
Step 1 :
Equation at the end of step 1 :
Step 2 :
x4 - 16
Simplify —————————————————
x3 + 2x2 + 4x + 8
Checking for a perfect cube :
2.1 x3 + 2x2 + 4x + 8 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3 + 2x2 + 4x + 8
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 4x + 8
Group 2: 2x2 + x3
Pull out from each group separately :
Group 1: (x + 2) • (4)
Group 2: (x + 2) • (x2)
-------------------
Add up the two groups :
(x + 2) • (x2 + 4)
Which is the desired factorization
Trying to factor as a Difference of Squares :
2.3 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 :
2.4 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 :
2.5 Factoring: x2 - 4
Check : 4 is the square of 2
Check : x2 is the square of x1
Factorization is : (x + 2) • (x - 2)
Polynomial Roots Calculator :
2.6 Find roots (zeroes) of : F(x) = x2 + 4
See theory in step 2.4
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
Canceling Out :
2.7 Cancel out (x2 + 4) which appears on both sides of the fraction line.
Canceling Out :
2.8 Cancel out (x + 2) which appears on both sides of the fraction line.
Final result :
x - 2
Its a quite long soln dude....