Step by Step Solution
Step 1 :
4
Simplify —
z
Equation at the end of step 1 :
4
(((z2) + 3z) - —) + 4
z
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using z as the denominator :
z2 + 3z (z2 + 3z) • z
z2 + 3z = ——————— = —————————————
1 z
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
z2 + 3z = z • (z + 3)
Adding fractions that have a common denominator :
3.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
z • (z+3) • z - (4) z3 + 3z2 - 4
——————————————————— = ————————————
z z
Equation at the end of step 3 :
(z3 + 3z2 - 4)
—————————————— + 4
z
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using z as the denominator :
4 4 • z
4 = — = —————
1 z
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(z) = z3 + 3z2 - 4
Polynomial Roots Calculator is a set of methods aimed at finding values of z for which F(z)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers z 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 | -2.00 | ||||||
| -2 | 1 | -2.00 | 0.00 | z + 2 | |||||
| -4 | 1 | -4.00 | -20.00 | ||||||
| 1 | 1 | 1.00 | 0.00 | z - 1 | |||||
| 2 | 1 | 2.00 | 16.00 | ||||||
| 4 | 1 | 4.00 | 108.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
z3 + 3z2 - 4
can be divided by 2 different polynomials,including by z - 1
Polynomial Long Division :
4.3 Polynomial Long Division
Dividing : z3 + 3z2 - 4
("Dividend")
By : z - 1 ("Divisor")
| dividend | z3 | + | 3z2 | - | 4 | ||||
| - divisor | * z2 | z3 | - | z2 | |||||
| remainder | 4z2 | - | 4 | ||||||
| - divisor | * 4z1 | 4z2 | - | 4z | |||||
| remainder | 4z | - | 4 | ||||||
| - divisor | * 4z0 | 4z | - | 4 | |||||
| remainder | 0 |
Quotient : z2+4z+4 Remainder: 0
Trying to factor by splitting the middle term
4.4 Factoring z2+4z+4
The first term is, z2 its coefficient is 1 .
The middle term is, +4z its coefficient is 4 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 4 .
| -4 | + | -1 | = | -5 | ||
| -2 | + | -2 | = | -4 | ||
| -1 | + | -4 | = | -5 | ||
| 1 | + | 4 | = | 5 | ||
| 2 | + | 2 | = | 4 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 2
z2 + 2z + 2z + 4
Step-4 : Add up the first 2 terms, pulling out like factors :
z • (z+2)
Add up the last 2 terms, pulling out common factors :
2 • (z+2)
Step-5 : Add up the four terms of step 4 :
(z+2) • (z+2)
Which is the desired factorization
Multiplying Exponential Expressions :
4.5 Multiply (z+2) by (z+2)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (z+2) and the exponents are :
1 , as (z+2) is the same number as (z+2)1
and 1 , as (z+2) is the same number as (z+2)1
The product is therefore, (z+2)(1+1) = (z+2)2
Adding fractions that have a common denominator :
4.6 Adding up the two equivalent fractions
(z+2)2 • (z-1) + 4 • z z3 + 3z2 + 4z - 4
—————————————————————— = —————————————————
z z
Checking for a perfect cube :
4.7 z3 + 3z2 + 4z - 4 is not a perfect cube
Trying to factor by pulling out :
4.8 Factoring: z3 + 3z2 + 4z - 4
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: z3 + 3z2
Group 2: 4z - 4
Pull out from each group separately :
Group 1: (z + 3) • (z2)
Group 2: (z - 1) • (4)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
4.9 Find roots (zeroes) of : F(z) = z3 + 3z2 + 4z - 4
See theory in step 4.2
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 | -6.00 | ||||||
| -2 | 1 | -2.00 | -8.00 | ||||||
| -4 | 1 | -4.00 | -36.00 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 2 | 1 | 2.00 | 24.00 | ||||||
| 4 | 1 | 4.00 | 124.00 |
Polynomial Roots Calculator found no rational roots
Final result :
z3 + 3z2 + 4z - 4
—————————————————
z
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