Problem 1 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) 3 terms: \(2x^5, 6x^4, 4x^3\). (b) Common factor: \(2x^3\). Factored: \(2x^3(x^2 + 3x + 2)\).

Step 1: Identify the Number of Terms

A term in a polynomial is each distinct part that may contain variables raised to powers, and have coefficients. In this polynomial, the terms are separated by addition signs. Here, we have the terms: \(2x^5\), \(6x^4\), and \(4x^3\). Thus, the polynomial has 3 terms.

Step 2: List the Terms

List each term from the polynomial separately. The terms of the polynomial \(2x^5 + 6x^4 + 4x^3\) are: - \(2x^5\)- \(6x^4\)- \(4x^3\).

Step 3: Find the Common Factor

To determine the common factor, find the greatest common divisor (GCD) of the coefficients and the lowest power of any common variables. For \(2x^5\), \(6x^4\), and \(4x^3\), the coefficients are 2, 6, and 4. The GCD is 2. Each term also contains at least \(x^3\). Thus, the common factor is \(2x^3\).

Step 4: Factor the Polynomial

To factor the polynomial \(2x^5 + 6x^4 + 4x^3\), divide each term by the common factor determined in Step 3, which is \(2x^3\):- \(2x^5 \div 2x^3 = x^2\)- \(6x^4 \div 2x^3 = 3x\)- \(4x^3 \div 2x^3 = 2\)Thus, the factored form is \(2x^3(x^2 + 3x + 2)\).

Key Concepts

Terms of a Polynomial

Polynomials are composed of terms, which are unique components separated by addition or subtraction signs. Each term can include constants, variables, and exponents. In the given polynomial \(2x^5 + 6x^4 + 4x^3\), there are three distinct terms.
- \(2x^5\)
- \(6x^4\)
- \(4x^3\)
Each term contains a coefficient (like 2, 6, or 4), a variable (\(x\)), and possibly an exponent (5, 4, or 3). Recognizing terms is crucial as it helps in identifying and understanding the structure and behavior of the polynomial in question.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of simpler polynomials or factors. This process usually requires identifying common factors in the terms of the polynomial.
In our example \(2x^5 + 6x^4 + 4x^3\), the goal is to find a factor that is common to all the terms. Breaking down into basics requires us to first determine the Greatest Common Divisor (GCD) among the terms.
Once identified, we divide each term in the polynomial by this common factor, effectively "factoring out" the common element. The factored polynomial for the example is \(2x^3(x^2 + 3x + 2)\), simplifying the expression and potentially making it easier to solve or analyze in further steps.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) in the context of polynomials is the highest degree of shared term among coefficients and variables in each term.
To find the GCD of the polynomial \(2x^5 + 6x^4 + 4x^3\), start with examining the coefficients (2, 6, and 4). The GCD of these numbers is 2. For the variables, each term contains \(x\) with the lowest power being \(x^3\).
Thus, the GCD of the terms includes both the numerical and variable components, specifically \(2x^3\). Using this GCD, we can simplify the polynomial by factoring it, leading to expressions that are frequently easier to use or apply in solving mathematical problems.