Problem 13 - College Algebra

Question

For the following exercises, factor by grouping. $$ 6 n^{2}-19 n-11 $$

Step-by-Step Solution

Answer

Factor: \((2n + 1)(3n - 11)\)

Step 1: Identify Terms to Group

Given the quadratic expression \(6n^2 - 19n - 11\), we want to rewrite it in a way that allows for easy factoring by grouping. Start by identifying pairs of terms that can be grouped together.

Step 2: Find Two Numbers

We need two numbers whose product is \(6 \times -11 = -66\) and whose sum is the middle coefficient, \(-19\). These numbers are \(-22\) and \(+3\).

Step 3: Rewrite the Expression

Rewrite the expression by breaking the middle term, \(-19n\), into \(-22n + 3n\), so it becomes \( 6n^2 - 22n + 3n - 11 \).

Step 4: Group the Terms

Group the terms into two pairs: \((6n^2 - 22n) + (3n - 11)\).

Step 5: Factor Each Group

Factor out the greatest common factor from each group. From \(6n^2 - 22n\), factor out \(2n\) to get \(2n(3n - 11)\). From \(3n - 11\), the greatest common factor is 1, so it remains \(1(3n - 11)\).

Step 6: Factor out the Common Term

Notice that both grouped terms now have a common factor of \((3n - 11)\). Factor \((3n - 11)\) out of the expression: \((2n + 1)(3n - 11)\).

Step 7: Verify the Solution

Multiply \((2n + 1)(3n - 11)\) to ensure it simplifies back to the original expression \(6n^2 - 19n - 11\). The solution is verified if it matches the original problem.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial of degree two. The general form is given by \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. In this exercise, our quadratic expression is \(6n^2 - 19n - 11\). This means the expression is based on the power of two as the highest power of the variable \(n\).

• Degree: The highest power of the variable, which is 2 in this case.
• Coefficients: The numerical factors of the terms, which are 6, -19, and -11 respectively.

Quadratic expressions can often be transformed into a product of two binomials through factoring techniques like factoring by grouping. This transformation makes it easier to solve equations where the expression is set equal to zero.

Common Factor

A common factor in polynomial terms is a number or variable that divides each term without a remainder. When factoring by grouping, finding a common factor within groups of terms is essential. In this exercise, each group needs a separate analysis to identify its specific common factors.

• First Group: \(6n^2 - 22n\), where the common factor is \(2n\).
• Second Group: \(3n - 11\), where the common factor is simply 1.

Finding the common factor allows us to break down the polynomial into simpler components, making it easier to see the expression in terms of its constituent parts. This step is crucial as it helps in finalizing the polynomial into a product of simpler expressions.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into a product of its simplest elements. It helps in simplifying expressions and solving equations. Factoring by grouping is a method of polynomial factoring particularly useful for quadratics.

• Step 1: Identify the quadratic expression and rewrite it if necessary to facilitate grouping.
• Step 2: Use grouping to organize terms into pairs based on shared factors.
• Step 3: Factor out the greatest common factor from each group.

In this particular case, we reorganized \(6n^2 - 19n - 11\) into two groups: \(6n^2 - 22n\) and \(3n - 11\). Then, we factored each to get \(2n(3n - 11)\) and \(1(3n - 11)\), which led us to the final factored form \((2n + 1)(3n - 11)\). This method often works well for quadratic expressions beyond the simple cases addressed by the quadratic formula.

Algebra Steps

Understanding and executing algebra steps is critical when factoring polynomial expressions. Let's break down the steps used in this solution:

• Step 1: Recognize which terms in the quadratic can be grouped together. This involves strategic planning based on the structure of the expression.
• Step 2: Solve for a pair of numbers that fit specific arithmetic conditions—multiplying to the product of the leading and constant coefficients and summing to the middle coefficient.
• Step 3: Rearrange the original expression, splitting the middle term into two parts based on the identified numbers.
• Step 4: Group terms accordingly, making sure each group can be individually factored.
• Step 5: Factor each group to reveal a common shape or term.
• Step 6: Carefully extract the common factor from the groups to achieve full factorization.
• Step 7: Confirm the factorization by multiplication, ensuring the original expression is perfectly reconstructed.

Completing these steps successfully requires attention to detail and practice. Each step should flow into the next, ultimately resulting in a completely factored expression.