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Chapter 5: Problem 47

Factor completely. $$ 4 a-5 a^{2}-10+2 a^{3} $$

Short Answer

Expert verified
(2a - 5)(a^{2} + 2)

Step by step solution

01

- Rearrange the terms

Rearrange the terms in descending order of their exponents.the given expression is:eq(2 a^{3} - 5 a^{2} + 4 a - 10)
02

- Group the terms

Group the terms to identify common factors.group as follows:eq(2 a^{3} - 5 a^{2}) + (4 a - 10)
03

- Factor out the GCF from each group

Factor the Greatest Common Factor (GCF) from each group separately.For eq(2 a^{3} - 5 a^{2}): eq a^{2}(2a - 5)For eq(4 a - 10): eq 2(2a - 5)
04

- Factor out the common binomial

Now, factor out the common binomial term eq(2a - 5) from both groups.The expression becomes:eq a^{2}(2a - 5) + 2(2a - 5)= eq(2a - 5)(a^{2} + 2)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rearranging Terms
Rearranging terms in a polynomial is often the first step in simplifying or factoring it. This process involves placing the terms in a specific order, usually by descending powers of a variable to make it easier to identify patterns. For example, the expression \(4a - 5a^{2} - 10 + 2a^{3}\) should be rearranged as \(2a^{3} - 5a^{2} + 4a - 10\) . This step sets a clearer path for further manipulation and reveals useful patterns.
Grouping
The next crucial step in factoring is grouping terms. Grouping helps in identifying common factors within small, manageable segments of the polynomial. Take our previous expression, \(2a^{3} - 5a^{2} + 4a - 10\). We can group the terms as follows:

  • \( (2a^{3} - 5a^{2}) + (4a - 10) \)
This grouping divides the polynomial into parts that can be factored independently, simplifying the process of finding common factors.
Greatest Common Factor
Finding the Greatest Common Factor (GCF) for each grouped segment is the next step. The GCF is the highest factor that divides each term within the group without leaving a remainder. Consider our grouped expression: \( (2a^{3} - 5a^{2}) + (4a - 10) \). In the first group \(2a^{3} - 5a^{2}\) , the GCF is \(a^{2}\). For the second group \(4a - 10\), the GCF is \(2\). Factoring out these GCFs gives us:

  • \( a^{2}(2a - 5) \)
  • \( 2(2a - 5) \)
This simplifies the polynomial further and prepares it for the final step of factoring.
Binomial Factorization
The final step is binomial factorization, where we factor out the common binomial from the grouped terms. For our simplified expression from previous sections, \(a^{2}(2a - 5) + 2(2a - 5)\) , we see that \(2a - 5\) is a common binomial term. Factoring out the common binomial gives us:

  • \( (2a - 5)(a^{2} + 2) \)
This expression is now fully factored. Each term in the polynomial has been successfully simplified into its binomial components, certifying that the original polynomial \( 4a - 5a^{2} - 10 + 2a^{3}\) has been completely factored.

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