Problem 24 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$7 x^{2}+28$$

Step-by-Step Solution

Answer

The factored form is \(7(x^2 + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, check for a common monomial factor in all the terms of the polynomial. The terms are \(7x^2\) and 28. Both numbers, 7 and 28, have a common factor of 7. Also, notice that there is no common variable factor between these two terms.

Step 2: Factor Out the GCF

Since 7 is the greatest common factor, factor it out from each term. This means we rewrite the polynomial \(7x^2 + 28\) as:\[7(x^2 + 4)\]

Step 3: Check for Further Factorization

Now, look at the expression inside the parentheses (\(x^2 + 4\)). Check if it can be factored further using integer coefficients. \(x^2 + 4\) is a sum of a square and a constant that cannot be factored further using integers.

Key Concepts

Greatest Common Factor (GCF)

When dealing with polynomial factorization, one of the first steps is to identify the Greatest Common Factor (GCF). This is a fundamental concept that helps simplify expressions and make calculations easier. The GCF is the largest factor shared by all the terms in a polynomial.

To determine the GCF:

• List the factors of each numerical coefficient.
• Identify the largest factor common to all of these lists.
• If there are variables, factor out any common variables shared by the terms with the lowest power.

For example, in the polynomial expression \(7x^2 + 28\), both terms have 7 as the numerical GCF. There are no common variables in this expression since only one term involves the variable \(x\).

By finding and factoring out the GCF, you simplify the polynomial to make it easier to work with in subsequent steps of factorization.

Integer Coefficients

Integers are whole numbers that do not include fractions or decimals. In polynomial factorization, integer coefficients refer to the numbers in front of the variables that are whole numbers. Each term in a polynomial can have coefficients, which are crucial when trying to factor the polynomial completely.

When working with polynomials like \(7x^2 + 28\), both 7 and 28 are integer coefficients. Factoring processes typically begin with whole numbers for easy manipulation. Check after any factorization process to see if the resulting expressions or terms can continue to be expressed with integer coefficients. This is important for maintaining the integrity of your factorization and ensuring accurate results.

Some polynomial expressions cannot be factored completely with integer coefficients, which means that their factors would involve non-whole numbers. In such cases, specify that the polynomial is not factorable over the integers.

Common Monomial Factor

The concept of a Common Monomial Factor is crucial in the initial step of polynomial factorization. A monomial factor is a single term consisting of the product of numbers and variables. A common monomial factor is then a monomial shared by all terms of a polynomial.

Identifying a common monomial factor involves:

• Finding the GCF of the numerical coefficients.
• Checking for common variable factors across terms.

Once identified, the common monomial factor is factored out of the polynomial, leading to a simpler expression inside the parentheses. For the polynomial \(7x^2 + 28\), the common monomial factor is 7.

Factoring out the common monomial factor simplifies the polynomial to \(7(x^2 + 4)\), making further examination easier. If a polynomial is expressed as the product of its common monomial factor and another polynomial that cannot be further factored with integer coefficients, then factorization is complete.