Problem 22 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(25 n^{2}-5 n-6\)

Step-by-Step Solution

Answer

Factored form is \((5n + 2)(5n - 3)\).

Step 1: Identify a, b, and c in the trinomial

The given trinomial is in the form of a quadratic expression \(ax^2 + bx + c\). Here, \(a = 25\), \(b = -5\), and \(c = -6\).

Step 2: Find two numbers that multiply to ac and add to b

Multiply \(a\) and \(c\): \(25 \times -6 = -150\). We need two numbers that multiply to \(-150\) and add to \(-5\). These numbers are \(-15\) and \(10\) because \(-15 \times 10 = -150\) and \(-15 + 10 = -5\).

Step 3: Rewrite the middle term using the numbers from Step 2

Rewrite \(-5n\) as \(-15n + 10n\). Now the expression becomes \(25n^2 - 15n + 10n - 6\).

Step 4: Factor by grouping

Group the terms: \((25n^2 - 15n) + (10n - 6)\). Factor out the greatest common factor from each group.\((25n^2 - 15n) = 5n(5n - 3)\) and \((10n - 6) = 2(5n - 3)\).

Step 5: Factor out the common binomial factor

Both groups contain the common factor \((5n - 3)\). Factor it out: \(5n(5n - 3) + 2(5n - 3) = (5n + 2)(5n - 3)\).

Step 6: Verification

Expand \((5n + 2)(5n - 3)\) to verify the solution: \(5n \times 5n = 25n^2\), \(5n \times -3 = -15n\), \(2 \times 5n = 10n\), \(2 \times -3 = -6\). Combining these gives: \(25n^2 - 15n + 10n - 6\), which matches the original expression.

Key Concepts

Quadratic Expressions

Quadratic expressions are algebraic expressions that represent polynomials of degree two. These have the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Quadratics can form parabolic graphs when plotted on a coordinate plane. They are fundamental in algebra because they often appear in mathematical problems and practical applications.Understanding quadratic expressions is crucial for solving a variety of problems in algebra, such as finding roots using the quadratic formula, completing the square, and more. In the exercise, we worked with the expression \(25n^2 - 5n - 6\), which is a classic example of a quadratic form, showcasing how to manipulate and factor these expressions efficiently.

Factoring by Grouping

Factoring by grouping is a technique used to factor polynomials by rearranging terms and identifying common factors. It's particularly helpful when dealing with trinomials that resist simple factorization.

Steps in Factoring by Grouping

In the given problem, we start by identifying the terms: \(25n^2 - 15n + 10n - 6\). We group these into two pairs: \( (25n^2 - 15n) \) and \( (10n - 6) \), making it easier to factor.

• Find the greatest common factor (GCF) in each group.
• The first group, \(25n^2 - 15n\), has a GCF of \(5n\).
• The second group, \(10n - 6\), has a GCF of \(2\).

By factoring these, we get expressions like \(5n(5n - 3)\) and \(2(5n - 3)\). Noticing the common binomial factor \((5n - 3)\) in both parts allows us to apply the factorization: \((5n + 2)(5n - 3)\). This method simplifies complex expressions into understandable terms, highlighting the power of group factoring.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two or more polynomial expressions. It's essential when verifying factored forms or in operations such as expanding expressions.Here, we verify the factorization result by expanding \((5n + 2)(5n - 3)\). This involves distributing each term in the first polynomial across each term in the second:

• \(5n \times 5n = 25n^2\)
• \(5n \times -3 = -15n\)
• \(2 \times 5n = 10n\)
• \(2 \times -3 = -6\)

Combining the results gives \(25n^2 - 15n + 10n - 6\). This matches our original trinomial, confirming that the factorization is accurate. Polynomial multiplication is thus an integral tool not only for verifying solutions but also for understanding the interrelations between factored and expanded forms.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. They are used to represent relationships and operations in a symbolic form. An algebraic expression can be anything from a simple single-term expression, like \(x\), to complex multi-term expressions like our given trinomial \(25n^2 - 5n - 6\).Understanding these expressions is fundamental to algebra, as they form the building blocks for almost all algebraic operations. When dealing with an algebraic expression like our quadratic trinomial, the key is to manipulate it into a simpler, more digestible form, such as its factored state.

The Role of Variables

Variables, like \(n\) in \(25n^2 - 5n - 6\), play a critical role as placeholders, allowing algebraic expressions to model real-world scenarios and represent general cases in mathematical problems. This symbolic representation is what makes algebraic expressions versatile and highly useful in problem-solving scenarios across various fields.