Problem 64 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 48-20 n+2 n^{2} $$

Step-by-Step Solution

Answer

The completely factored form is \(2(n-6)(n-4)\).

Step 1: Identify the GCF

To factor the trinomial, first identify the greatest common factor (GCF) of all the terms. The terms are 48, -20n, and 2n². The GCF is 2, as it evenly divides each term.

Step 2: Factor out the GCF

Divide each term by the GCF and factor it out of the trinomial. The expression becomes:\[2(24 - 10n + n^2)\]

Step 3: Rearrange and Simplify

Rearrange the trinomial inside the parenthesis into standard quadratic form. This gives us:\[2(n^2 - 10n + 24)\]

Step 4: Factor the Quadratic

To factor the quadratic \(n^2 - 10n + 24\), look for two numbers whose product is 24 (the constant term) and sum is -10 (the coefficient of the linear term). These numbers are -6 and -4.

Step 5: Write the Factorization

Use the numbers found in Step 4 to express the quadratic as a product of two binomials:\[2(n - 6)(n - 4)\]

Key Concepts

Greatest Common Factor

When you start factoring any trinomial, it's essential to first identify the greatest common factor (GCF). The GCF is the largest number that evenly divides each term of the expression. In the case of the trinomial \(48 - 20n + 2n^2\), you look for a number that all three terms have in common. Here, the GCF is 2. Recognizing the GCF simplifies the expression and makes it easier to factor the remaining parts.

• Identify each coefficient or constant in the expression.
• Determine the largest number that divides all these terms completely without leaving a remainder.
• Extract or "factor out" this number as a common factor from each term.

By factoring out the GCF, you simplify the trinomial into a form that is easier to work with. In our example, dividing each term by 2 gives you \(2(24 - 10n + n^2)\). This crucial step reduces errors in the later stages of factorization.

Quadratic Factorization

Quadratic factorization follows after simplifying the trinomial by removing the GCF. A quadratic trinomial generally takes the form \(ax^2 + bx + c\). The goal is to rewrite this expression as the product of two binomials. This involves a few steps:

• Identify the "a", "b", and "c" terms, which correspond to the coefficients of \(n^2\), \(n\), and the constant term.
• Look for two numbers that multiply to "c" and add to "b".
• Rearrange into binomials: \((x + m)(x + n)\), where "m" and "n" are the numbers found.

For example, in \(n^2 - 10n + 24\), simplify further by finding two numbers whose product is 24 and sum is -10. The numbers -6 and -4 meet these conditions, leading to \((n - 6)(n - 4)\). It helps visualize the relationship between factor pairs of "c" and solution methods used to derive quadratic binomials.

Algebraic Expressions

An understanding of algebraic expressions is fundamental when working with polynomials. These expressions consist of variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division. In this context, they help in representing complex relationships through simple mathematical forms.

• Algebraic expressions showcase relationships and patterns in functions and equations.
• They can be manipulated using operations to simplify or to switch to a factor form.
• Identifying terms and coefficients helps to manage expressions effectively.

For example, the expression \(2(n^2 - 10n + 24)\) involves understanding how the symbols and numbers interact. By rearranging and simplifying, you uncover deeper patterns that aid in solving the problem.

Polynomial Simplification

Simplifying polynomials is crucial because it reduces them to their most basic form, making them easier to understand and solve. For polynomials, simplification often involves factoring out any common factors and rewriting the polynomial as the product of simpler expressions.

• Identify and remove the greatest common factor.
• Rewrite expressions to facilitate easier manipulation and understanding.
• Factor the polynomial into simpler terms or expressions.

In this exercise, the trinomial \(48 - 20n + 2n^2\) was simplified by identifying the GCF of 2, rearranging into a standard quadratic formula, and finally expressing it as \(2(n - 6)(n - 4)\). The simplified expression reveals solutions and demonstrates how complex problems, when broken down, become manageable and comprehensible.