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Chapter 5: Problem 163

Factor each polynomial completely. $$a^{5}+4 a^{3}$$

Short Answer

Expert verified
The factored form is \(a^{3}(a^{2} + 4)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

The given polynomial is \(a^{5} + 4a^{3}\). The first step is to find the greatest common factor (GCF) of the terms. Both terms have a common factor of \(a^{3}\).
02

Factor Out the GCF

Factor out \(a^{3}\) from both terms in the polynomial. This gives \(a^{3}(a^{2} + 4)\).
03

Check for Further Factorization

After factoring out the GCF, the resulting polynomial is \(a^{3}(a^{2} + 4)\). Check if \(a^{2} + 4\) can be factored further. Since \(a^{2} + 4\) cannot be factored further (it is not a difference of squares and has no real roots), the factorization is complete.

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Key Concepts

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

greatest common factor
When factoring polynomials, one of the first steps is to find the greatest common factor (GCF). The GCF is the largest factor that divides all the terms in the polynomial.
For example, in the polynomial \(a^{5} + 4a^{3}\), both terms contain the factor \(a^{3}\). Therefore, \(a^{3}\) is the GCF. This means we can factor out \(a^{3}\) from each term.
By factoring out the GCF, the polynomial simplifies and becomes easier to work with. It helps break down complex polynomials into simpler parts. Finding the GCF requires identifying the lowest power of common variables or numbers shared by all terms.
step-by-step solution in algebra
Breaking down complex algebra problems step-by-step makes them easier to understand. Here鈥檚 how you can apply this method:
  • First, identify the problem. For example, the polynomial \(a^{5} + 4a^{3}\).
  • Next, find the greatest common factor (GCF) of the terms. In this case, the GCF is \(a^{3}\).
  • Factor out the GCF from the polynomial: \(a^{3}(a^{2} + 4)\).
  • Then, check if the resulting polynomial can be factored further. For \(a^{2} + 4\), it cannot be simplified further.
  • The final factored form is \(a^{3}(a^{2} + 4)\).
This method of breaking down the problem step-by-step ensures clarity and makes complex problems more manageable.
factoring polynomials
Factoring polynomials is a key skill in algebra. It involves expressing a polynomial as a product of simpler polynomials that, when multiplied, give you the original polynomial. Here鈥檚 how it works:
  • Find the GCF of the terms. For example, in \(a^{5} + 4a^{3}\), the GCF is \(a^{3}\).
  • Factor out the GCF: \(a^{3}(a^{2} + 4)\).
  • Determine if the remaining polynomial can be factored further. \(a^{2} + 4\) cannot be factored further because it doesn't fit into patterns like the difference of squares.
Factoring simplifies complex polynomials and is essential for solving polynomial equations. It鈥檚 important to practice recognizing different factoring patterns and techniques, such as factoring by grouping, using the difference of squares, and quadratic trinomials. In our example, after factoring out the GCF and finding no further factorizable elements, the simplified polynomial is left as \(a^{3}(a^{2} + 4)\).

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