Problem 29 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 x^{5}+2 x^{2}$$

Step-by-Step Solution

Answer

The complete factorization of the given polynomial \(2 x^{5}+2 x^{2}\) is \(2 x^{2}(x^{3}+1)\)

Step 1: Identify Common Factors

The first step in factorization usually involves identifying any common factors across all terms in the polynomial. In the given polynomial \(2 x^{5}+2 x^{2}\), \(2 x^{2}\) is a common factor as it appears in both terms.

Step 2: Factor the Polynomial

The next step is to factor the polynomial by dividing each term of the polynomial by the common factor \(2 x^{2}\). This results in the factored form: \(2 x^{2}(x^{3}+1)\). The polynomial is now completely factored, as no other factors can be extracted from the polynomial.

Step 3: Check the Factorization

Lastly, check your factorization by expanding your factored polynomial to see if it matches with the original one. When expanded, \(2 x^{2}(x^{3}+1)\) gives the original polynomial \(2 x^{5}+2 x^{2}\). Hence, the factorization is correct.

Key Concepts

common factor

When tackling the factorization of a polynomial, identifying the **common factor** is usually the first step. The common factor refers to a number or expression that is present in each term of a polynomial expression. For instance, consider the polynomial given in the exercise:

• Terms: \(2x^5\) and \(2x^2\)
• Common Factor: \(2x^2\)

This means you are looking for the largest term or expression that each part of the polynomial shares.
By factoring this common term out from each term of the polynomial, you simplify it into a more useful expression.
Remember, factoring out the greatest common factor is always your starting point when aiming to factor completely.

factoring techniques

In algebra, using various **factoring techniques** helps in breaking down complex polynomials into simpler parts. After identifying the common factor, further methods could be used if needed.

• **Common Factoring**: Identifying and extracting the common factors, as seen in our original polynomial \(2x^5+2x^2\).
• **Grouping**: This is used when dealing with polynomials with four or more terms, by grouping terms that have a common factor.
• **Difference of Squares and Trinomials**: Special techniques used for certain structured polynomials, although they aren't applicable in this specific example.

In this exercise, applying the common factoring technique was sufficient to achieve complete factorization. By recognizing and pulling out \(2x^2\), we decomposed the polynomial into \(2x^2(x^3+1)\).
This reveals the structure of the polynomial and makes further operations, like multiplying back to check your work, much easier.

algebraic expressions

An **algebraic expression** is a mathematical phrase that can include numbers, variables, and operation symbols. Understanding how to manipulate and simplify these expressions is key in algebra. A polynomial like the one in the exercise is a specific type of algebraic expression.

• **Polynomial**: A sum of terms consisting of a coefficient, a variable raised to a whole number exponent, or both.
• **Terms**: In \(2x^5 + 2x^2\), each part separated by a '+' or '-' is called a term.
• **Coefficient**: The numerical part in front of the variables, such as '2' in this exercise.

By identifying these elements, you gain insight into the structure of the expression and how different techniques like factoring can be applied to simplify or solve it.
Remember, the simplicity in algebraic expressions often leads to easier problem-solving in broader algebraic contexts.