Problem 92 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 t^{2}-100$$

Step-by-Step Solution

Answer

The polynomial factors to \(25(t + 2)(t - 2)\).

Step 1: Identify Common Factor

The polynomial given is \(25t^2 - 100\). First, let's identify any common factors in both terms. The common factor here is 25.

Step 2: Factoring Out the Common Factor

Take the common factor, which is 25, out of the polynomial.\[25t^2 - 100 = 25(t^2 - 4)\]

Step 3: Recognize the Difference of Squares

The expression inside the parenthesis \(t^2 - 4\) is a difference of squares, which can be factored further. Recall the formula: \(a^2 - b^2 = (a+b)(a-b)\).

Step 4: Apply the Difference of Squares Formula

In the expression \(t^2 - 4\), identify \(a = t\) and \(b = 2\). Factor it using the difference of squares formula: \[(t^2 - 4) = (t + 2)(t - 2)\]

Step 5: Write the Completely Factored Form

Thus, the expression \(25(t^2 - 4)\) becomes:\[25(t + 2)(t - 2)\] This is the completely factored form of the polynomial.

Key Concepts

Common Factor

When beginning to factor polynomials, one of the first and essential steps is identifying the common factor. A common factor is a number or expression that can evenly divide each term of the polynomial.
In the polynomial \(25t^2 - 100\), the number 25 can be seen in both \(25t^2\) and 100. This makes 25 the greatest common factor.

• To factor out the common factor, you divide each term of the polynomial by 25.
• This means separating the common factor from each term to simplify the expression.

By factoring out the greatest common factor, the polynomial is reduced, making it easier to tackle further factoring techniques.
After removing the common factor, our expression becomes \(25(t^2 - 4)\). This step simplifies the polynomial and sets it up for further factoring down the line.

Difference of Squares

The difference of squares is a special type of factorization. It applies to expressions that are made up of two perfect squares separated by a subtraction sign. The general formula is: \[a^2 - b^2 = (a+b)(a-b)\]In this formula, each of the squares is represented by \(a^2\) and \(b^2\).
Tensorial to our polynomial \(25(t^2 - 4)\), we see that inside the parenthesis, \(t^2 - 4\) is a simple difference of squares. Identifying \(t^2\) as \(a^2\) and \(4\) as \(b^2\) (since \(4\) can be written as \(2^2\)).
This reveals the setup for utilizing the difference of squares:

• \(a = t\)
• \(b = 2\)

Apply the difference of squares formula to transform \(t^2 - 4\) into \((t + 2)(t - 2)\). Recognizing and applying this powerful tool allows for efficient and complete polynomial factoring.

Factoring Techniques

Factoring techniques combine various strategies to break down polynomials. By employing different methods, such as finding the common factor or recognizing patterns like the difference of squares, complex polynomials become manageable.

Steps to Factor Thoroughly:

• **Identify Perfect Squares:** Look for terms that can be written as squares.
• **Find the Common Factor:** Always seek a number or expression present in all polynomial's terms and factor it out.
• **Apply Formulae:** Knowing formulas, like the difference of squares or trinomials, speeds up the process.
• **Check Your Work:** After breaking it down, verify by multiplying the factors to ensure you reach the initial polynomial.

Using these techniques, expressions like \(25t^2 - 100\) can be broken down step by step. Thus, the completely factored form ends up being \(25(t + 2)(t - 2)\). Becoming familiar with these methods simplifies many algebraic challenges.