Answer :
Final Answer:
The expression 48x³ - 20x² - 8x factors to 4x(3x - 2)(4x + 1).
Explanation:
To factor the given expression 48x³ - 20x² - 8x completely, we can first look for the greatest common factor (GCF) among the terms. In this case, the GCF is 4x, so we can factor it out:
4x(12x² - 5x - 2).
Now, we need to factor the quadratic expression 12x² - 5x - 2. To do this, we look for two numbers that multiply to the leading coefficient (12) times the constant term (-2) and add up to the middle coefficient (-5).
The two numbers that fit these criteria are -6 and 1 because (-6) * 1 = 12 and (-6) + 1 = -5. So, we can use these numbers to factor the quadratic:
4x(12x² - 6x + 1x - 2).
Next, we group the terms and factor by grouping:
4x(6x(2x - 1) + 1(2x - 1)).
Now, we can see that we have a common factor of (2x - 1) in both terms, so we factor that out:
4x(2x - 1)(6x + 1).
So, the expression 48x³ - 20x² - 8x factors completely to 4x(3x - 2)(4x + 1). Each of these factors is irreducible.
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