Problem 15 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 15 x^{2}+11 x+2 $$

Step-by-Step Solution

Answer

The trinomial \(15x^2 + 11x + 2\) factors into \((5x + 2)(3x + 1)\).

Step 1: Identify a × c

To factor the trinomial by grouping, first identify the coefficient of the first term (\(a\)) and the constant term (\(c\)). Here, \(a = 15\) and \(c = 2\). Multiply them: \(a \, \times c = 15 \, \times \, 2 = 30\).

Step 2: Find two numbers that multiply to a × c

Find two numbers that multiply to 30 (the product of \(a\) and \(c\)) and add up to the middle term coefficient, which is 11. The numbers 5 and 6 satisfy this condition because \(5 \times 6 = 30\) and \(5 + 6 = 11\).

Step 3: Rewrite the middle term using these numbers

Rewrite the trinomial as four terms, using the two numbers identified: \(15x^2 + 5x + 6x + 2\). This does not change the value of the expression but allows us to factor by grouping.

Step 4: Group the terms

Group the expression into two pairs: \((15x^2 + 5x) + (6x + 2)\).

Step 5: Factor each group

Factor out the greatest common factor from each group. The first group is \(15x^2 + 5x\), and the GCF is \(5x\), resulting in \(5x(3x + 1)\). In the second group, \(6x + 2\), the GCF is 2, resulting in \(2(3x + 1)\).

Step 6: Factor by grouping

Notice both terms \((5x(3x+1) + 2(3x+1))\) contain a common binomial factor \((3x + 1)\). Factor this out to get \((5x + 2)(3x + 1)\).

Key Concepts

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler polynomials whose product equals the original polynomial. This technique is particularly useful for simplifying expressions and solving polynomial equations.
The process involves finding polynomial terms that, when multiplied together, reproduce the original polynomial. Here's a basic outline:

• Identify common factors in the polynomial terms.
• Decompose the original polynomial into more manageable parts.
• Utilize techniques like the grouping method or special formulas.

By factoring polynomials, you can easily manipulate and analyze algebraic expressions. It's a cornerstone of algebra, helping with everything from solving equations to analyzing geometric shapes.

Grouping Method

The grouping method is an efficient technique for factoring polynomials, especially helpful for trinomials. It works by reorganizing the polynomial into smaller, more manageable parts. Here's how it generally functions:

• Identify terms that can be grouped together based on common factors.
• Factor out the greatest common factor from each group.
• Look for a common binomial in the results, and factor it out.

The grouping method simplifies the polynomial step-by-step, which can be particularly beneficial when the polynomial is complex, such as trinomial expressions with various coefficients. This method enhances clarity and delivers a straightforward path to the polynomial’s factors.

Trinomial Expressions

Trinomial expressions are polynomials that consist of three terms, typically in the form of ax² + bx + c. They are common in algebra because they often represent quadratic equations. Key points of understanding trinomial expressions include:

• The leading term is the one with the highest degree; usually, this is the squared term ax².
• The middle term bx can be broken down to assist in factoring, often using methods like the grouping method.
• The constant term c affects the vertex and roots of the polynomial if it’s part of a quadratic equation.

Recognizing how to manipulate each part of a trinomial is crucial for effective factoring. The ability to rewrite and transform a trinomial is a valuable skill that unlocks further algebraic processes, including solving quadratic equations.

Quadratic Equations

Quadratic equations are a special type of polynomial equation where the highest power of the variable is 2. They typically take the form ax² + bx + c = 0. Solving these involves finding values of x that satisfy the equation, and factoring is one method to find these solutions.
To solve quadratic equations by factoring:

• First, express the quadratic as a trinomial.
• Use factoring techniques like the grouping method to break down the trinomial.
• Set each factor to zero to find the roots of the equation.

This method is especially useful when the quadratic can be factored easily, resulting in straightforward solutions. Quadratic equations appear frequently in various mathematical and real-world contexts, making mastering their solutions an essential algebraic skill.