Problem 1 - Precalculus Mathematics for Calculus

Question

Consider the polynomial \(2 x^{5}+6 x^{4}+4 x^{3}\) How many terms does this polynomial have?___ List the terms:____ What factor is common to each term?______ Factor the polynomial: \(2 x^{5}+6 x^{4}+4 x^{3}=\)_____

Step-by-Step Solution

Answer

The polynomial has 3 terms. The common factor is \(2x^3\). Factored form: \(2x^3(x^2 + 3x + 2)\).

Step 1: Count the Terms

The polynomial given is \(2x^5 + 6x^4 + 4x^3\). Count the number of distinct terms in the polynomial. These are the expressions separated by plus signs. Therefore, this polynomial has three terms.

Step 2: List the Terms

The terms of the polynomial \(2x^5 + 6x^4 + 4x^3\) are: \(2x^5, 6x^4, 4x^3\).

Step 3: Identify the Common Factor

Examine each term to find the greatest common factor. Each term has a factor of 2 and at least \(x^3\). Thus, the greatest common factor common to all terms is \(2x^3\).

Step 4: Factor the Polynomial

To factor the polynomial, divide each term by the common factor \(2x^3\) and then factor this common factor out. - For \(2x^5\), factoring out \(2x^3\) gives: \(x^2\).- For \(6x^4\), factoring out \(2x^3\) gives: \(3x\).- For \(4x^3\), factoring out \(2x^3\) gives: \(2\).Thus, the factored polynomial is: \(2x^3(x^2 + 3x + 2)\).

Key Concepts

Understanding the Greatest Common Factor (GCF)

When dealing with polynomials, finding the greatest common factor (GCF) is a crucial step in simplification and factoring processes. The GCF is the highest factor that divides each term of a polynomial completely. It is especially important because it simplifies the polynomial and makes it easier to manipulate.

Here is how you can determine the GCF:

• Identify the numerical coefficients and variable parts of each term in the polynomial.
• Find the common factors of these coefficients.
• Identify the lowest power of variables common to each term.
• The product of these factors is your GCF.

For example, in the polynomial \(2x^5 + 6x^4 + 4x^3\), each term has a number and a variable component. The numbers 2, 6, and 4 share a greatest numerical factor of 2. For the variables, the smallest power common to each is \(x^3\). Thus, the GCF is \(2x^3\). By factoring this out, you're essentially expressing the polynomial in a simpler form, such as \(2x^3(x^2 + 3x + 2)\).

Identifying Polynomial Terms

Polynomials are made up of terms, each consisting of a coefficient, a variable, and an exponent. Understanding these terms is key to working with polynomials effectively. In mathematical expressions, terms are usually separated by addition or subtraction signs.

Consider the polynomial \(2x^5 + 6x^4 + 4x^3\). This has three distinct terms:

• \(2x^5\)
• \(6x^4\)
• \(4x^3\)

Each term is a separate entity within the polynomial expression. The term "polynomial" itself indicates "many terms," inherently showing the diversity and complexity these expressions can have.

Normally, when asked to "list the terms," it means breaking down the polynomial and identifying each distinct expression that provides its complete description. This simplification helps in recognizing how each part contributes to the whole expression.

Breaking Down the Factored Polynomial

Factoring a polynomial means expressing it as a product of its simplest components. This process simplifies the polynomial and can make solving equations easier. The first step in this process is to factor out the greatest common factor.

For the polynomial \(2x^5 + 6x^4 + 4x^3\):

• Find the GCF, which we determined is \(2x^3\).
• Divide each term by the GCF to find the remaining factors.

After factoring, we have:

• \((2x^5) \div (2x^3) = x^2\)
• \((6x^4) \div (2x^3) = 3x\)
• \((4x^3) \div (2x^3) = 2\)

So, the original polynomial simplifies into the factored form: \(2x^3(x^2 + 3x + 2)\). This transformation reveals the simplified structure of the polynomial, providing an easier pathway for further calculations, such as solving for specific variable values or graphing.

Diving into Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Polynomials are a specific type of algebraic expression that incorporates variables raised to whole number powers.

In contrast to simple expressions like \(7 + 3\), polynomials such as \(2x^5 + 6x^4 + 4x^3\) demonstrate the power of algebra in expressing complex relationships. These expressions are powerful tools in encapsulating a wide variety of real-world scenarios and mathematical concepts.

Understanding each component of algebraic expressions helps to manipulate and solve many types of algebra problems. Variables act as placeholders that can represent numbers. Coefficients provide the multiplicative power of variables, while exponents signify the level of the polynomial. Being comfortable with these elements allows for easy factorization and simplification, crucial steps in algebra.