Final Answer:
The factors of the given expression are x(x² - 7)(x² + 3).
Step-by-step explanation:
To factor the given expression completely, we start by looking for the greatest common factor (GCF) and any patterns that might help us simplify the expression. The expression given is:
x⁵ - 21x - 4x³
Let's first rearrange the terms in descending order of the powers of x:
x⁵ - 4x³ - 21x
Now let's try and see if there is any common factor in all terms.
Each term contains an x, so we can factor x out:
x(x⁴ - 4x² - 21)
Now we are left with the expression:
x⁴ - 4x² - 21
This is a quadratic in form with respect to x², so we can use the techniques applicable for factoring quadratics.
We need to find two numbers that multiply to -21 (the constant term) and add to -4 (the coefficient of the x² term).
These numbers are -7 and +3.
Now we can express the quadratic part as:
x⁴ - 7x² + 3x² - 21
Next, we group the terms:
(x⁴ - 7x²) + (3x² - 21)
We can factor out x² from the first group and 3 from the second group:
x²(x² - 7) + 3(x² - 7)
Now, we have a common factor of (x² - 7):
(x² - 7)(x² + 3)
Our completely factored expression is then:
x(x² - 7)(x^2 + 3)
This is the completely factored form of the original expression.