Problem 73 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 4 t^{2}+36 $$

Step-by-Step Solution

Answer

The expression factors to \( 4(t^2 + 9) \).

Step 1: Identify Common Factors

Look for any common factors in the expression. The expression is \( 4t^2 + 36 \). Notice that both terms are divisible by 4. This is a common factor.

Step 2: Factor Out the Common Number

Factor out the greatest common factor, which is 4. This gives us: \( 4(t^2 + 9) \).

Step 3: Assess the Remaining Expression

Examine the expression inside the parentheses, \( t^2 + 9 \). Determine if it can be further factored. \( t^2 + 9 \) is a sum of squares, which is not factorable over the real numbers.

Step 4: Determine if Expression is Prime

Since no other common factors are found and \( t^2 + 9 \) is not factorable as a sum of squares, the expression \( 4(t^2 + 9) \) is fully factored.

Key Concepts

Greatest Common Factor

When factoring expressions, the first step often involves finding the greatest common factor (GCF). The GCF is the largest number that can evenly divide all terms in the expression. For the expression \(4t^2 + 36\), we find that both terms, \(4t^2\) and \(36\), are divisible by 4. This means 4 is the GCF.
By factoring out this 4, we are simplifying the expression to \(4(t^2 + 9)\). This makes further assessment and operations easier. The GCF is a foundational step that sets up the process for additional factoring or identifying special kinds of expressions.
Remember, always look for a GCF first, as it can greatly simplify the remainder of the problem.

Sum of Squares

The expression left inside the parentheses after factoring the GCF is \(t^2 + 9\). This represents what we call a 'sum of squares', where two square terms are summed together.
Unlike the difference of squares, which follows a clear factorization pattern, the sum of squares does not factor over the real numbers in a simple way. This means that \(t^2 + 9\) generally remains unchanged in terms of further factorization using real numbers.
It's important to recognize this kind of expression because attempting to factor it as you would with a difference of squares will not work. However, recognizing it as a sum alerts you that it is already in its simplest form for real number operations.

Prime Expression

In mathematics, a 'prime expression' is an expression that cannot be factored into simpler polynomial expressions with integer coefficients. For the given expression \(4(t^2 + 9)\), we have simplified it as much as possible once we have factored out the GCF and identified the sum of squares.
Since \(t^2 + 9\) cannot be further factored over the real numbers into simpler terms, it is considered a "prime" expression. This term is akin to prime numbers that only have 1 and themselves as factors, indicating the indivisibility of the expression towards further simple factorization.
Knowing when an expression is considered prime is essential for determining when to stop attempting further factorization and acknowledging you have reached its most reduced form.