Problem 52 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 5 n^{2}-10 n-10 $$

Step-by-Step Solution

Answer

The completely factored form is \(5(n^2 - 2n - 2)\), and the polynomial \(n^2 - 2n - 2\) is prime.

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor in all the terms of the polynomial. The given polynomial is \(5n^2 - 10n - 10\). The GCF of the coefficients (5, 10, 10) is 5. So, factor out 5.

Step 2: Factor out the GCF

Factor out the GCF, which is 5, from each term:\[5n^2 - 10n - 10 = 5(n^2 - 2n - 2)\].

Step 3: Check the Quadratic Polynomial

Look inside the parentheses at the quadratic polynomial \(n^2 - 2n - 2\). Check if this quadratic can be factored further. To factor it, we need two numbers that multiply to -2 (the constant term) and add to -2 (the coefficient of the linear term).

Step 4: Attempt to Find Factors

Try to find pairs of numbers that multiply to -2 and add to -2. They are: (1, -2) and (-2, 1), none of which meet the criteria. Therefore, \(n^2 - 2n - 2\) cannot be factored further with integer coefficients.

Step 5: Declare the Polynomial Prime

Since \(n^2 - 2n - 2\) cannot be factored further, it is considered a prime polynomial.

Step 6: Write the Final Answer

The completely factored form of the polynomial is \(5(n^2 - 2n - 2)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides each of the terms in a polynomial without leaving a remainder. To find the GCF, identify all the factors of each term. For example:

• The terms in the polynomial \(5n^2 - 10n - 10\) are \(5n^2\), \(-10n\), and \(-10\).
• Factor out each of these terms. The common factor in all these terms is 5.

Once you find the GCF, factor it out from the polynomial. In the given example, factoring out 5 gives us \(5(n^2 - 2n - 2)\). This step simplifies the polynomial and makes it easier to work with in further steps.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored further using integers. In simpler terms, if no combination of factors can simplify the polynomial without leaving a remainder, it is prime.
For example:

• In the polynomial \(n^2 - 2n - 2\), check if it can be factored anymore.
• Look for two numbers that multiply to the constant term (-2) and add to the coefficient of the linear term (-2). No such numbers exist.

Because you cannot find those numbers, \(n^2 - 2n - 2\) is a prime polynomial. Hence, in the original exercise, we end up with \(5(n^2 - 2n - 2)\) as the final answer, where \(n^2 - 2n - 2\) is a prime polynomial.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2, usually in the form \(ax^2 + bx + c\). These types of polynomials can often be factored or simplified. To determine if a quadratic polynomial can be factored, follow these steps:

• Identify the coefficients: In \(n^2 - 2n - 2\), \(a=1\), \(b=-2\), and \(c=-2\).
• Look for two numbers that multiply to the constant term (\(c=-2\)) and add to the linear coefficient (\(b=-2\)).
• If such numbers exist, factor the quadratic. If not, declare it prime.

In the given example, we attempted to factor \(n^2 - 2n - 2\) and determined that no pairs of integers meet the criteria. Hence, we concluded that it is a prime polynomial. But the general process involves checking for these pairs and factoring them if they exist.