Factor the polynomial completely
x^3-4x^2-21x
To factor the polynomial x^3 - 4x^2 - 21x completely, we can first look for common factors. In this case, we can factor out an x:
x(x^2 - 4x - 21)
Next, we can further factor the quadratic expression inside the parentheses. We look for two numbers that multiply to give -21 and add up to -4. The numbers -7 and 3 satisfy these conditions:
x(x - 7)(x + 3)
Therefore, the polynomial x^3 - 4x^2 - 21x factors completely as x(x - 7)(x + 3).
can you show all the steps
Sure! Here are all the steps to factor the polynomial x^3 - 4x^2 - 21x completely:
1. Look for common factors: We can factor out an x from each term:
x(x^2 - 4x - 21)
2. Factor the quadratic expression: We look for two numbers that multiply to give -21 and add up to -4. The numbers -7 and 3 satisfy these conditions, so we can write the quadratic as:
x(x - 7)(x + 3)
3. Combine all the factors: Putting it all together, the polynomial factors completely as:
x(x - 7)(x + 3)