Factor polynomial $$ 20x^{3}-96x^{2}-144x $$
$$ 20x^{3}-96x^{2}-144x = 4x( x-6 )( 5x+6 ) $$
Step 1 :
After factoring out $ 4x $ we have:
$$ 20x^{3}-96x^{2}-144x = 4x ( 5x^{2}-24x-36 ) $$Step 2 :
Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:
Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -36} $.
$$ a \cdot c = -180 $$Step 4: Find out two numbers that multiply to $ a \cdot c = -180 $ and add to $ b = -24 $.
Step 5: All pairs of numbers with a product of $ -180 $ are:
| PRODUCT = -180 | |
| -1 180 | 1 -180 |
| -2 90 | 2 -90 |
| -3 60 | 3 -60 |
| -4 45 | 4 -45 |
| -5 36 | 5 -36 |
| -6 30 | 6 -30 |
| -9 20 | 9 -20 |
| -10 18 | 10 -18 |
| -12 15 | 12 -15 |
Step 6: Find out which factor pair sums up to $\color{blue}{ b = -24 }$
| PRODUCT = -180 and SUM = -24 | |
| -1 180 | 1 -180 |
| -2 90 | 2 -90 |
| -3 60 | 3 -60 |
| -4 45 | 4 -45 |
| -5 36 | 5 -36 |
| -6 30 | 6 -30 |
| -9 20 | 9 -20 |
| -10 18 | 10 -18 |
| -12 15 | 12 -15 |
Step 7: Replace middle term $ -24 x $ with $ 6x-30x $:
$$ 5x^{2}-24x-36 = 5x^{2}+6x-30x-36 $$Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -6 $ out of the last two terms.
$$ 5x^{2}+6x-30x-36 = x\left(5x+6\right) -6\left(5x+6\right) = \left(x-6\right) \left(5x+6\right) $$