Problem 18 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ 2 n^{2}-28 n+98 $$

Step-by-Step Solution

Answer

The completely factored form is \( 2(n - 7)^2 \).

Step 1: Identify the Common Factor

First, look for a common factor in all the terms of the trinomial. Each term in the trinomial, \(2n^2\), \(-28n\), and \(98\), is divisible by 2. So, factor out the \(2\) from the entire trinomial to get: \[ 2(n^2 - 14n + 49) \]

Step 2: Recognize a Perfect Square Trinomial

The expression inside the parentheses \(n^2 - 14n + 49\) is a perfect square trinomial. Recall that a perfect square trinomial \(a^2 - 2ab + b^2\) can be factored as \((a-b)^2\). Here, \(a = n\) and \(b = 7\) because \(7^2 = 49\) and \(-2 \times 7 = -14\). Thus, rewrite the expression as a square: \[ n^2 - 14n + 49 = (n - 7)^2 \]

Step 3: Write the Final Factored Form

Substitute the factored form of the trinomial it back into the expression with the common factor. Thus, the completely factored form of the original trinomial is: \[ 2(n - 7)^2 \] This means that after factoring out the common factor and recognizing the trinomial as a perfect square, the expression is now fully factored.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a polynomial that can be expressed in the form \(a^2\), \(-2ab\), \(b^2\), and is denoted mathematically as \(a^2 - 2ab + b^2 = (a-b)^2\). To identify a perfect square trinomial, check if the first term and the last term are perfect squares themselves. Then, confirm that the middle term is twice the product of the square roots of the first and last terms. Perfect square trinomials emerge often in algebra, and recognizing them helps simplify complex polynomial expressions.

• Look for patterns like this to factor trinomials quickly and efficiently.
• In our exercise, \(n^2 - 14n + 49\) fits this pattern perfectly, because \(n\) is the square root of \(n^2\) and \(7\) is the square root of \(49\).

Once you rewrite the trinomial as \((n - 7)^2\), you've effectively used the properties of perfect square trinomials for simplification and better understanding of the polynomial structure.

Common Factor

A common factor is an element that divides each term of a given polynomial without leaving any remainder. Identifying and factoring out the common factor is an essential first step in polynomial simplification. This process not only simplifies the expression but also prepares it for further operations, like recognizing special polynomial forms.

• Identify the greatest common factor in all terms, which maximizes the simplification.
• In our trinomial \(2n^2 - 28n + 98\), each term is divisible by \(2\). Thus, \(2\) is our common factor.

After factoring out the \(2\), we are left with \(n^2 - 14n + 49\), swiftly simplifying the problem and making it ready for step two, which involves checking for a perfect square trinomial.

Polynomial Expressions

Polynomial expressions are algebraic terms involving sums of powers of variables with coefficients. They can range from simple expressions like \(x^2 + 2x + 1\) to more complex formulas with multiple variables and varied powers. Understanding how to work with polynomial expressions is a foundational skill in algebra.

• The expressions can often be simplified or factored into basic components.
• This helps in solving equations or simplifying expressions for easier interpretation.

Trinomials like \(2n^2 - 28n + 98\) are polynomial expressions with three terms. The goal when working with such trinomials is often to factor them completely, as seen in the exercise. Knowing that a polynomial can be rewritten through factoring aids in solving equations and unlocking the underlying algebraic structure.