Questions: Factor completely. 10x^3 - 81x^2 - 81x

Transcript text: Factor completely. \[ 10 x^{3}-81 x^{2}-81 x \]

Solution

Solution Steps

To factor the given polynomial completely, we can start by factoring out the greatest common factor (GCF) from all the terms. Then, we can look for patterns or use methods such as grouping or the quadratic formula to factor the remaining polynomial.

Step 1: Factor Out the GCF

The given polynomial is \( 10x^3 - 81x^2 - 81x \). We start by factoring out the greatest common factor (GCF), which is \( x \). This gives us: \[ x(10x^2 - 81x - 81) \]

Step 2: Factor the Quadratic

Next, we need to factor the quadratic \( 10x^2 - 81x - 81 \). We can look for two numbers that multiply to \( 10 \times (-81) = -810 \) and add to \( -81 \). The quadratic can be factored as: \[ 10x^2 - 81x - 81 = (x - 9)(10x + 9) \]

Step 3: Combine the Factors

Now, we combine the factors we found: \[ 10x^3 - 81x^2 - 81x = x(x - 9)(10x + 9) \]