Question

Factor z 3 - 2z 2 + 9z - 18.

Answer 1

z3-2z2+9z-18

Final result :

(z2 + 9) • (z - 2)
Reformatting the input :

Changes made to your input should not affect the solution:

(1): "z2" was replaced by "z^2". 1 more similar replacement(s).

Step by step solution :

Step 1 :

Raise z to the 2nd power
Exponentiation :

Equation at the end of step 1 :

(((z3) - (2 • z2)) + 9z) - 18
Step 2 :

Raise z to the 3rd power
Exponentiation :

Equation at the end of step 2 :

((z3 - 2z2) + 9z) - 18
Step 3 :

Simplify z3-2z2+9z - 18
Checking for a perfect cube :

3.1 z3-2z2+9z-18 is not a perfect cube

Trying to factor by pulling out :

3.2 Factoring: z3-2z2+9z-18

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: 9z-18
Group 2: z3-2z2

Pull out from each group separately :

Group 1: (z-2) • (9)
Group 2: (z-2) • (z2)
-------------------
Add up the two groups :
(z-2) • (z2+9)
Which is the desired factorization

Polynomial Roots Calculator :

3.3 Find roots (zeroes) of : F(z) = z2+9
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 9.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 10.00
-3 1 -3.00 18.00
-9 1 -9.00 90.00
1 1 1.00 10.00
3 1 3.00 18.00
9 1 9.00 90.00

Polynomial Roots Calculator found no rational roots

Final result :

(z2 + 9) • (z - 2)

Answer 2

z^2(z-2)+9(z-2)=(z^2+9)(z-2)=(z+3i)(z-3i)(z-2)

Answer 3

http://www.tiger-algebra.com/drill/z3-2z2_9z-18/