Factor the expression completely: 9x^2 - 24x^5
Understand the Problem
The question is asking us to factor the given polynomial expression, which involves identifying common factors and rewriting the expression in a multiplied form.
Answer
The completely factored form of the expression is $$ 3x^2(3 - 8x^3). $$
Answer for screen readers
The completely factored form of the expression is $$ 3x^2(3 - 8x^3). $$
Steps to Solve
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Identify common factors
First, we look for common numerical coefficients and variable factors in the expression (9x^2 - 24x^5). The coefficients are 9 and 24. The greatest common divisor (GCD) of 9 and 24 is 3. The variable part is (x^2) (the lowest power of (x) in the expression). -
Factor out the GCD
Now we will factor out the GCD, which is (3x^2): $$ 9x^2 - 24x^5 = 3x^2(3 - 8x^3) $$ -
Identify further factorization
Next, we check if the expression (3 - 8x^3) can be factored further. In this case, (3 - 8x^3) does not factor nicely over the real numbers, as it is neither a difference of squares nor a sum/difference of cubes. -
Write the completely factored form
The completely factored form of the expression is: $$ 3x^2(3 - 8x^3) $$
The completely factored form of the expression is $$ 3x^2(3 - 8x^3). $$
More Information
Factoring polynomials is a fundamental skill in algebra. It allows us to simplify expressions and solve equations more easily. The technique of factoring out the greatest common factor is commonly used in polynomial expressions.
Tips
- Neglecting to factor completely: Make sure to check if the remaining polynomial can be factored further. In this case, (3 - 8x^3) cannot be factored further.
- Forgetting to include the variable when factoring: Ensure that you factor out the lowest power of any common variables.
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