Problem 19 - 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. $$7 x^{3}+7 x$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(7x^{3}+7x\) is \(7x(x^{2} + 1)\).

Step 1: Identifying common factors

Looking at our polynomial \(7x^{3}+7x\), we can observe that 7 and x are common to both terms. We therefore can factor out the greatest common factor, which is \(7x\). This gives us an expression as \(7x(x^{2} + 1)\).

Step 2: Further Factorization

Looking at the expression \(x^{2} + 1\) that we now have, it cannot be factorized any further using real numbers as it involves the sum of squares (not the difference of squares).

Step 3: Checking the factorization

To confirm our factorization you could multiply \(7x\) and \((x^{2} + 1)\) using the distributive law. This produces \(7x^{3} + 7x\) which matches our original polynomial.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by two or more algebraic terms. To find the GCF, we look for the highest exponent of shared variables and the highest divisible number among the coefficients. In the exercise provided, we have the polynomial \(7x^{3}+7x\). Both terms share the number 7 and the variable x to at least the first power. Consequently, the GCF is \(7x\), which we can factor out. By extracting the GCF from the polynomial, the complexity of the expression is reduced, making the algebraic manipulation and simplification processes easier.

For students, understanding the concept of the GCF involves recognizing patterns and commonalities within algebraic expressions, which is a fundamental skill in algebra that can greatly simplify solving equations and understanding mathematical relationships.

Sum of Squares

The sum of squares is a concept where two squaring terms are added together, resulting in the form \(a^{2} + b^{2}\). This form is notably different from the difference of squares, which is \(a^{2} - b^{2}\) and can be factored into \((a+b)(a-b)\). However, the sum of squares, such as the expression \(x^{2} + 1\) from our problem, cannot be factored into a simpler form using real numbers. This is because it does not fit into any patterns that allow for factorization, unlike the difference of squares which can be easily factored due to its inherent pattern. The sum of squares often represents a stopping point for factorization when dealing with real numbers.

When attempting to factor an algebraic expression, recognizing a sum of squares can save students time and confusion by making it clear that further factorization, using real numbers, is not possible.

Distributive Law

The distributive law, also known as the distributive property, is a fundamental rule in algebra that allows us to multiply a single term by each term within a parenthesis. The general form of the distributive law is \(a(b + c) = ab + ac\). When applied to the context of our exercise, after factoring out the GCF and obtaining \(7x(x^{2} + 1)\), we could reverse the process to check our work. By applying the distributive law, we multiply \(7x\) by each term inside the parentheses, \(x^{2}\) and 1, which gives us the original polynomial, \(7x^{3}+7x\).

Understanding and correctly applying the distributive law is essential for students, as it allows not only for simplifying expressions but also for verifying factorizations by 're-multiplying' the factors to check if the original expression is obtained.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They do not contain equals signs, which distinguishes them from equations. In algebraic expressions, like the one in our exercise, understanding how to combine like terms, factor common factors, and apply the distributive law are key to simplifying the expression. An important aspect of algebraic expressions is the ability to manipulate them in various ways, such as factorization, to either simplify the expression or solve for a variable.

Algebraic expressions are the building blocks for more complex mathematical concepts; hence, a solid grasp of how to work with them, including all associated properties and rules, is pivotal for any student's success in mathematics.