To factor the polynomial \( 20x^3 - 96x^2 - 144x \), let's first factor out the greatest common factor (GCF).
The GCF of the coefficients \( 20, -96, \) and \( -144 \) is \( 4 \). Additionally, all terms have an \( x \) factor. The GCF is therefore \( 4x \).
We can factor \( 4x \) out of the polynomial:
\[ 20x^3 - 96x^2 - 144x = 4x(5x^2 - 24x - 36) \]
Next, we need to factor the quadratic \( 5x^2 - 24x - 36 \). To do this, we can use the AC method or look for two numbers that multiply to \( 5 \cdot (-36) = -180 \) and add up to \( -24 \).
The two numbers that fit this requirement are \( -30 \) and \( 6 \), since:
\[ -30 + 6 = -24 \quad \text{and} \quad -30 \cdot 6 = -180 \]
Now we can rewrite the quadratic:
\[ 5x^2 - 30x + 6x - 36 \]
Next, we group the terms:
\[ (5x^2 - 30x) + (6x - 36) \]
Factoring out the common factors in each group:
\[ 5x(x - 6) + 6(x - 6) \]
Now we can factor out \( (x - 6) \):
\[ (5x + 6)(x - 6) \]
Putting it all together, we have:
\[ 20x^3 - 96x^2 - 144x = 4x(5x + 6)(x - 6) \]
Thus, the linear factors of the polynomial \( 20x^3 - 96x^2 - 144x \) are:
\[ 4x, \quad 5x + 6, \quad \text{and} \quad x - 6 \]
1 answer