Problem 33 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(18 x^{2}-9 x-14\)

Step-by-Step Solution

Answer

The trinomial factors as \((6x - 7)(3x + 2)\).

Step 1: Identify the Coefficients

For the trinomial \(18x^2 - 9x - 14\), identify the coefficients: \(a = 18\), \(b = -9\), and \(c = -14\).

Step 2: Find Two Numbers that Multiply to ac

Calculate \(ac = 18 \times (-14) = -252\). We need two numbers that multiply to \(-252\) and add to \(-9\), the coefficient of the \(x\)-term.

Step 3: Determine Suitable Pair of Numbers

The suitable pair of numbers is \(-21\) and \(12\) because \((-21) + 12 = -9\) and \((-21) \times 12 = -252\).

Step 4: Rewrite the Middle Term

Rewrite \(-9x\) as \(-21x + 12x\) in the trinomial: \(18x^2 - 21x + 12x - 14\).

Step 5: Factor by Grouping

Group terms: \((18x^2 - 21x) + (12x - 14)\).

Step 6: Factor Each Group

Factor out the greatest common factor from each group: \(3x(6x - 7) + 2(6x - 7)\).

Step 7: Factor Out the Common Binomial Factor

The common factor in both terms is \((6x - 7)\), so the factored expression is \((6x - 7)(3x + 2)\).

Step 8: Verify the Factorization

Expand \((6x - 7)(3x + 2)\) to verify: \[ (6x - 7)(3x + 2) = 18x^2 + 12x - 21x - 14 = 18x^2 - 9x - 14 \]The factorization is correct.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is 2. This results in an expression of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic equations appear in various scenarios, such as physics problems, optimization issues, and geometry.

• They can be solved in multiple ways, including factoring, completing the square, and using the quadratic formula.
• These equations have a parabolic graph and can open upwards or downwards depending on the sign of \(a\).

Understanding quadratic equations lays the foundation for solving and factoring different polynomial expressions effectively.

Factoring by Grouping

Factoring by grouping is a useful technique to factor more complex polynomial expressions, such as trinomials. The process begins by rearranging the terms in a way that they can be grouped to find common factors.
Here's how it typically works:

• Identify pairs of terms that can be grouped together to reveal a common factor.
• Extract the common factor from each group of terms, simplifying the expression.
• Look for a common binomial factor in the new simplified expression, then factor it out completely.

In the context of the original trinomial \(18x^2 - 9x - 14\), grouping effectively helped in breaking down the expression into factorable portions, leading to the final factored form.

Polynomial Expressions

Polynomial expressions consist of terms, each containing a variable raised to a non-negative integer power, multiplied by a coefficient. These expressions form the basis of many algebraic problems.
In general, polynomial expressions can be simple or quite intricate with multiple terms, but they all follow the same basic structure:

• Each term is structured as \(ax^n\), where \(a\) is a coefficient and \(n\) is a whole number.
• Polynomials can be combined, added, or even multiplied to form new expressions.

When factoring algebraic polynomials, it is crucial to consider each term and its role in the overall expression to pinpoint the simplest form possible. The original trinomial exercise focuses on turning a complex polynomial expression into a simpler, factored one.

Algebraic Factorization

Algebraic factorization is the process of breaking down an expression into products of simpler expressions, or factors. This is particularly vital when working with complex trinomials or polynomial equations.
The benefits of factorization include:

• Simplifying complex expressions, making them easier to handle and understand.
• Identifying roots or solutions of polynomial equations easier, which is critical in solving equations.
• Providing insights into the structure and behavior of algebraic expressions.

For the trinomial \(18x^2 - 9x - 14\), successful factorization involved identifying factors that combine to reproduce the original expression, thereby verifying and simplifying the trinomial into \((6x - 7)(3x + 2)\). Factorization is essential in algebra as it opens up new methods to solve equations and analyze polynomial behavior.