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Chapter 6: Problem 4

Factor completely. Identify any prime polynomials. $$ x^{3}+5 x^{2}+4 x $$

Short Answer

Expert verified
Factor is \(x(x + 1)(x + 4)\). All components are fully factored and prime.

Step by step solution

01

Factor out the Greatest Common Factor (GCF)

Identify the greatest common factor in all terms of the polynomial. All terms in the polynomial have a common factor of x. Factor x out of the polynomial: \[ x(x^{2} + 5x + 4) \]
02

Factor the Quadratic Expression

Now, focus on the quadratic expression inside the parentheses: \(x^{2} + 5x + 4\). Look for two numbers that multiply to 4 (the constant term) and add up to 5 (the coefficient of the linear term x). These numbers are 1 and 4. Therefore, we can factor the quadratic as: \[ (x + 1)(x + 4) \]
03

Combine All Factors

Insert the factored quadratic back with the factored x from Step 1. Our fully factored polynomial is: \[ x(x + 1)(x + 4) \]
04

Identify Prime Polynomials

A prime polynomial cannot be factored into polynomials of lower degree with integer coefficients. In this case, we have factored the polynomial completely, and each factor is a linear polynomial which is prime.

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Key Concepts

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

Greatest Common Factor (GCF)
Let's start with the greatest common factor (GCF). The GCF is the largest factor that divides all terms of a polynomial. It helps in simplifying the polynomial by factoring out the common elements. Take our given polynomial:
\(x^{3} + 5x^2 + 4x\). Here, each term has at least one factor of \(x\). So, we can factor out \(x\) from each term:
  • \(x^3 = x \times x^2\)
  • \(5x^2 = x \times 5x\)
  • \(4x = x \times 4\)

Therefore, by factoring out the GCF, we get \(x(x^2 + 5x + 4)\). Now our polynomial is simpler and easier to handle.

Factoring Quadratic Expressions
Next, let鈥檚 focus on the quadratic expression inside the parentheses: \(x^2 + 5x + 4\). Factoring quadratic expressions typically involves finding two numbers that multiply to the constant term (in this case, 4) and add to the coefficient of the linear term (5). For our given quadratic, we look for two numbers that:
  • Multiply to 4
  • Add up to 5

These numbers are 1 and 4, because:
  • \(1 \times 4 = 4\)
  • \(1 + 4 = 5\)

This lets us factor the quadratic as \((x + 1)(x + 4)\). Now, by combining this with our previous step, the polynomial factors to \(x(x + 1)(x + 4)\). This fully simplifies our polynomial.
Prime Polynomials
Finally, let's discuss prime polynomials. A prime polynomial cannot be factored into lower-degree polynomials with integer coefficients. It's similar to a prime number in number theory, which only has 1 and itself as factors. In our polynomial, the factors are \(x\), \((x + 1)\), and \((x + 4)\). Each of these is a linear polynomial and cannot be factored any further using integer coefficients. So, they are considered prime polynomials. By recognizing prime polynomials, we understand that we have factored the polynomial as completely as possible.

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