Question

Find all of the roots of the following quartic equation: h(x)= x^4 - 2x^3 - 9x^2 + 18x

Answer 1

X4)-(2•(x3)))-32x2)+18x
STEP
2
:

Equation at the end of step
2
:

(((x4) - 2x3) - 32x2) + 18x
STEP
3
:

STEP
4
:
Pulling out like terms

4.1 Pull out like factors :

x4 - 2x3 - 9x2 + 18x =

x • (x3 - 2x2 - 9x + 18)

Checking for a perfect cube :

4.2 x3 - 2x2 - 9x + 18 is not a perfect cube

Trying to factor by pulling out :

4.3 Factoring: x3 - 2x2 - 9x + 18

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

Group 1: -9x + 18
Group 2: x3 - 2x2

Pull out from each group separately :

Group 1: (x - 2) • (-9)
Group 2: (x - 2) • (x2)
-------------------
Add up the two groups :
(x - 2) • (x2 - 9)
Which is the desired factorization

Trying to factor as a Difference of Squares:

4.4 Factoring: x2 - 9

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 : 9 is the square of 3
Check : x2 is the square of x1

Factorization is : (x + 3) • (x - 3)

Final result :
x • (x + 3) • (x - 3) • (x - 2)