Factoring
We want to factor the following polynomial:
18x³ - 51x² + 36x
First step
We find the Greatest Common Factors of its terms.
Since
3 · 6 = 18
3 · 17 = 51
3 · 12 = 36
we can rewrite the polynomial as:
18x³ - 51x² + 36x
↓
3 · 6x³ - 3 · 17x² + 3 · 12x
Then, the common factors are
x and 3
We group those factors:
3 · 6x³ - 3 · 17x² + 3 · 12x
↓
3x (6x² - 17x + 12)
Second step
Now, we have
3x (6x² - 17x + 12)
and we want to factorise
6x² - 17x + 12
We want to factor by Grouping
Then, we have to split the middle term -17x, so we can factor it
Since -17x = -8x - 9x
we are going to use it:
6x² - 17x + 12
↓
6x² - 8x - 9x + 12
Now, we group the first two terms together and then the last two terms together and factor by GCF:
6x² - 8x - 9x + 12
↓
(6x² - 8x) + (-9x + 12)
For the first bynomial
Since
2 · 3 = 6
2 · 4 = 8
The first polynomial can be rewritten as:
6x² - 8x = 2 · 3x² - 2 · 4x
It has in common 2 and x, then:
2 · 3x² - 2 · 4x = 2x (3x - 4)
For the second bynomial
Since
-3 · 3 = -9
-3 · (-4) = 12
The second bynomial can be rewritten as:
9x + 12 = -3 · 3x + -3 · (-4)
They have in common -3, then:
-9x + 12 = -3(3x - 4)
Third step
Now,we have that
6x² - 8x - 9x + 12
= 2x (3x - 4) - 3(3x - 4)
Then
3x(6x² - 8x - 9x + 12) = 3x[2x (3x - 4) - 3(3x - 4)]
We can see that
2x (3x - 4)
and
- 3(3x - 4)
have in common (3x - 4)
Then
2x (3x - 4) - 3(3x - 4)
= (3x - 4)(2x - 3)
Then, the whole polynomial can be written as
18x³ - 51x² + 36x
= 3x[2x (3x - 4) - 3(3x - 4)]
= 3x(3x - 4)(2x - 3)
Answer: 18x³ - 51x² + 36x = 3x(3x - 4)(2x - 3)