Problem 45 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ \frac{1}{3} x^{4}+\frac{2}{3} x^{3}-\frac{4}{3} x^{5}+\frac{1}{3} x $$

Step-by-Step Solution

Answer

The GCF is \(\frac{1}{3}x\); factored form is \(\frac{1}{3}x (-4x^4 + x^3 + 2x^2 + 1)\).

Step 1: Identify the GCF of the coefficients

The coefficients of the polynomial are \(\frac{1}{3}, \frac{2}{3}, -\frac{4}{3}, \text{ and } \frac{1}{3}\). The greatest common factor (GCF) of these coefficients is \(\frac{1}{3}\) since it is the largest coefficient that divides each of these numbers exactly.

Step 2: Identify the GCF of the variables

The variable terms are \(x^4, x^3, x^5, \text{ and } x\). The greatest common factor among these variable terms is \(x\) because it is the lowest power of \(x\) present in all terms.

Step 3: Factor out the GCF

Combine the GCF of the coefficients with the GCF of the variables to get \(\frac{1}{3}x\). Factor \(\frac{1}{3}x\) out of the polynomial: \[ \frac{1}{3}x ( x^3 + 2x^2 - 4x^4 + 1 ) \] This simplification leverages the GCF to rewrite the initial polynomial.

Step 4: Simplify the expression inside the parentheses

Rearrange and simplify the terms inside the parentheses: Start with \( -4x^4 + x^3 + 2x^2 + 1 \). Now arrange them by the descending power of \(x\): \[ \frac{1}{3}x ( -4x^4 + x^3 + 2x^2 + 1) \] The polynomial is already in its simplest form since no like terms can be combined further.

Key Concepts

Greatest Common Factor (GCF)

In mathematics, the Greatest Common Factor (GCF) helps simplify expressions, making calculations easier. The GCF of a set of numbers is the largest number that divides all numbers in the set without leaving a remainder.

Finding the GCF is crucial when working with polynomials. It helps simplify these expressions by determining a factor that can be pulled out of every term. To find the GCF for the coefficients in a polynomial, as demonstrated, identify the largest number that divides them all. In this exercise, the coefficients were fractional: \(\frac{1}{3}, \frac{2}{3}, -\frac{4}{3}, \text{ and } \frac{1}{3}\). Here, the GCF is \(\frac{1}{3}\), since

• It divides each coefficient exactly without a remainder, and
• It is the largest such divisor for these terms.

Identifying the GCF for variables involves finding the lowest power of a variable present in all terms, which leads us to the next concept.

Polynomial Expressions

Polynomial expressions are statements formed by adding, subtracting, or multiplying different terms. Each term usually consists of

• a coefficient
• one or more variables
• and a non-negative integer power of the variable(s).

In the given exercise, the polynomial is composed of four terms:

• \(\frac{1}{3}x^4\)
• \(\frac{2}{3}x^3\)
• -\(\frac{4}{3}x^5\), and
• \(\frac{1}{3}x\).

Understanding these expressions involves identifying similar terms and looking for common factors across them. Factoring these expressions can significantly simplify and illuminate the characteristics of the polynomial, like finding roots and improving ease of computation. Removing the GCF from polynomial expressions simplifies calculations and reveals locational structures such as symmetry and vertex forms in further exploration. In this exercise, factoring relies heavily on recognizing and using common factors. This process extends beyond just the coefficients to include powers of variables, which brings us to our third core aspect.

Powers of Variables

The power, or exponent, of a variable in a polynomial denotes how many times you'll multiply the variable by itself. For example, in the term \(x^4\), the variable \(x\) is raised to the fourth power, meaning \(x\times x\times x\times x\).

When factoring polynomials, the exponents are crucial as they determine the degree of the polynomial, impacting both algebraic operations and graph characteristics. The lowest power of a variable that's present in each term of a polynomial is key when identifying the GCF of variables. In this exercise, the powers present were \(x^4, x^3, x^5, \) and \(x\).

Since \(x\) is the smallest power in all terms, it becomes the variable part of the GCF, making the full GCF \(\frac{1}{3}x\). Simplifying involves factoring this GCF out from each term, reducing complexity and creating a straightforward expression. Recognizing the impact of variable powers enables smoother manipulation and understanding within a broader mathematical space.