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Chapter 5: Problem 38

Factor each polynomial. $$ x^{4}-4 x^{3} $$

Short Answer

Expert verified
The polynomial factors to \(x^3(x - 4)\).

Step by step solution

01

Factor out the Greatest Common Factor (GCF)

Identify the greatest common factor in each term of the polynomial. The terms are \(x^4\) and \(-4x^3\). Both terms have \(x^3\) in common. Factoring \(x^3\) out, the expression becomes \(x^3(x - 4)\).
02

Verify Factored Form

Distribute \(x^3\) back into the parentheses to ensure the factored expression matches the original polynomial. \(x^3(x - 4) = x^4 - 4x^3\). Since this is equivalent to the original polynomial, the factoring is verified as correct.

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Key Concepts

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

Greatest Common Factor
The greatest common factor, or GCF, is a key concept in simplifying and factoring polynomials. It is the largest factor that divides all the terms in a polynomial without leaving any remainder.
  • The GCF makes factoring easier because it simplifies the polynomial expression by reducing it to smaller, more manageable parts.
  • In polynomials, the GCF will always include the highest power of common variables across all terms.
For example, in the polynomial expression \(x^4 - 4x^3\), the terms are \(x^4\) and \(-4x^3\). Here, each term includes the variable \(x\) raised to some power, and the powers in common are \(x^3\). So, \(x^3\) is the GCF, simplifying the process of factoring out the common elements from the polynomial.
Polynomial Expressions
Polynomial expressions are a cornerstone of algebra, composed of terms that are added or subtracted together. Each term typically features a variable raised to a power, and it may also include a constant multiplied by the variable.
  • The polynomial \(x^4 - 4x^3\) is a simple example, where you have two terms: \(x^4\) and \( -4x^3\).
  • These terms make it a polynomial expression, where you focus on identifying common factors for factoring or further simplifications.
Understanding the structure of polynomials allows us to use factoring techniques more effectively and solve equations or simplify expressions more easily.
Algebraic Factoring Techniques
Algebraic factoring techniques simplify polynomial expressions by breaking them down into simpler forms. One common technique is to use the GCF, but there are various other methods to factor.
  • After identifying the GCF in a polynomial expression like \(x^4 - 4x^3\), we factor it out. This results in \(x^3(x - 4)\).
  • Factoring techniques continue beyond the GCF, involving methods like grouping, difference of squares, or quadratic factoring for more complex polynomials.
To ensure accuracy, it's crucial to verify by expanding the factored terms back to the original expression. In our exercise, distributing \(x^3\) across \((x - 4)\) confirms that \(x^3(x - 4)\) equals \(x^4 - 4x^3\). This ensures the polynomial is accurately factored.

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