Problem 23 - 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. $$5 x^{2}+5$$

Step-by-Step Solution

Answer

The polynomial factors as \(5(x^2 + 1)\).

Step 1: Identify the Common Factor

First, we look for any common monomial factor in the terms of the polynomial. Both terms in the expression \(5x^2 + 5\) have a factor of 5.

Step 2: Factor Out the Common Monomial Factor

We can factor out the common factor of 5, simplifying the expression to \(5(x^2 + 1)\).

Step 3: Check for Further Factorization

Next, examine the expression \(x^2 + 1\). It cannot be factored further over the integers, as it does not represent a difference of squares or any recognizably factorable pattern using integer coefficients.

Key Concepts

Common Monomial Factor

Factoring polynomials often begins with identifying the common monomial factor. A monomial is simply a single term, and when multiple terms in a polynomial share a common factor, we can factor it out to simplify the expression. In our example, examine the polynomial \(5x^2 + 5\). Here, both terms \(5x^2\) and \(5\) share a factor of 5.
This means that 5 is a common monomial factor, which we can extract from each term.
By factoring out 5, our polynomial simplifies to \(5(x^2 + 1)\).

• Look for any greatest common divisor in all terms.
• Factor it out to form a simpler expression.

This step is essential as it primes the polynomial for further examination and potential factorization.

Integers

When working with polynomials, it’s important to determine if the expression can be factored completely using integers. Integers are whole numbers, including positive, negative numbers, and zero.
They do not include fractions or decimals.
In our polynomial \(5(x^2 + 1)\), the expression inside the parentheses, \((x^2 + 1)\), needs to be checked for factorability over the integers.

• If a polynomial cannot be factored further using only integers, it is considered 'irreducible over the integers.'
• This means that the polynomial is already in its simplest form when using integer coefficients.

For \(x^2 + 1\), no integer factors exist, as it does not equate to any recognizable pattern that fits into basic factorization techniques with integers.

Difference of Squares

The 'difference of squares' is a specific pattern in algebra used for factorization. It refers to expressions that can be written in the form \(a^2 - b^2\), which can be factored as \((a + b)(a - b)\).
It’s a vital technique in simplifying polynomials.
If you spot a clear difference of squares, it can be factored directly using this formula.

• Recognize the pattern where two perfect squares are subtracted from each other.
• Apply the factorization formula \((a + b)(a - b)\).

In our example, \(x^2 + 1\) is not a difference of squares – note the plus sign. Thus, it cannot be factored further using this technique.

Factorization Techniques

Factorization involves breaking down an expression into products of simpler expressions, usually algebraic ones, that when multiplied, yield the original expression. Several techniques are employed:

• Prime Factorization: Breaking down numbers into their prime components.
• Common Monomial Factoring: Looking for shared factors across terms, as seen with \(5x^2 + 5\).
• Dfference of Squares: Recognizing and applying patterns to simplify expressions.
• Trinomials: Factoring expressions with three terms, often using the quadratic formula if necessary.

It’s crucial to apply the right technique for each specific polynomial.
In some cases, polynomials cannot be simplified using integer factorizations, indicating that they are irreducible over the integers.
Each step brings us closer to a concise and factorable form, enhancing our capacity to solve algebraic equations effectively.