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Chapter 6: Problem 3

Factor completely. Identify any prime polynomials. $$ x^{3}+6 x^{2}+5 x $$

Short Answer

Expert verified
The fully factored form is \( x(x + 1)(x + 5) \).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, look at all the terms in the polynomial to see if there is a common factor. Here, all terms have an 'x'. So, the GCF is 'x'.
02

Factor Out the GCF

Factor 'x' out of each term in the polynomial: \( x^3 + 6x^2 + 5x = x(x^2 + 6x + 5) \)
03

Factor the Trinomial

Now, focus on the trinomial \( x^2 + 6x + 5 \). Look for two numbers that multiply to 5 (the constant term) and add up to 6 (the coefficient of the middle term). These numbers are 1 and 5.
04

Write the Factors

Write the trinomial as the product of two binomials based on the numbers found in Step 3: \( x^2 + 6x + 5 = (x + 1)(x + 5) \)
05

Combine All Factors

Combine all the factored parts together: \( x(x^2 + 6x + 5) = x(x + 1)(x + 5) \)

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Key Concepts

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

Greatest Common Factor (GCF)
When factoring polynomials, it's often helpful to start by identifying the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in the polynomial. For example, in the polynomial \(x^3 + 6x^2 + 5x\), each term contains the variable 'x'. Therefore, 'x' is the GCF.

By factoring out the GCF, we simplify the polynomial into a more manageable form. After factoring out 'x', we get:
\[ x^3 + 6x^2 + 5x = x(x^2 + 6x + 5) \]
This operation reduces the polynomial to a product of the GCF and another polynomial, making further factorization easier.
Trinomial Factorization
After factoring out the GCF, you may often find yourself with a trinomial that needs further factorization. A trinomial is an algebraic expression with three terms, such as \(x^2 + 6x + 5\).
To factor a trinomial, we look for two numbers that multiply to give the constant term (in this case, 5) and add up to give the coefficient of the middle term (in this case, 6). Here, those two numbers are 1 and 5.

This lets us write the trinomial as a product of two binomials:
\[ x^2 + 6x + 5 = (x + 1)(x + 5) \]
This method is a key step in breaking down complex polynomials into simpler factors.
Prime Polynomial
Not all polynomials can be factored into simpler polynomials with integer coefficients. Those that can't be factored further are called 'prime polynomials'. Identifying a prime polynomial is crucial in polynomial factorization because it indicates that we've simplified the polynomial as much as possible.

In our example, after factoring out the GCF and the trinomial, we combined everything together:
\[ x(x^2 + 6x + 5) = x(x + 1)(x + 5) \]
The polynomial \(x + 1\) and \(x + 5\) can't be factored further, indicating that each factor is prime. This concludes the factorization process, and confirms we've broken down the original polynomial completely.

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