Problem 68 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(-15 x^{2}+26 x-8\)

Step-by-Step Solution

Answer

The trinomial \(-15x^2 + 26x - 8\) factors to \((5x + 2)(3x - 4)\).

Step 1: Identify the problem structure

The trinomial given is \(-15x^2 + 26x - 8\). Our task is to factor this trinomial completely. It follows the quadratic form \(ax^2 + bx + c\), where \(a = -15\), \(b = 26\), and \(c = -8\).

Step 2: Multiply and find product

Multiply the leading coefficient \(a\) and the constant term \(c\). Here, \((-15) imes (-8) = 120\). The product is \(120\). We will use this to find two numbers that multiply to \(120\) and add to \(26\).

Step 3: Find numbers that satisfy conditions

Look for two numbers that multiply to \(120\) and add up to the middle coefficient \(26\). These numbers are \(20\) and \(6\) because \(20 \times 6 = 120\) and \(20 + 6 = 26\).

Step 4: Break the middle term

Rewrite the middle term \(26x\) using \(20x\) and \(6x\) as follows: \(-15x^2 + 20x + 6x - 8\).

Step 5: Factor by grouping

Group the terms: \((-15x^2 + 20x) + (6x - 8)\). Factor out the greatest common factor from each group: \(5x(-3x + 4) + 2(3x - 4)\).

Step 6: Factor out common binomial

Notice that both groups contain the binomial \((3x - 4)\). Factor this common binomial from each group: \((5x + 2)(3x - 4)\). This is the fully factored form of the original trinomial.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in the form \(ax^2 + bx + c = 0\). They are called 'quadratic' because they involve terms squared, or raised to the power of two. In the context of factoring, we focus on expressions like \(ax^2 + bx + c\) instead of equations. Here, the goal is to rewrite the quadratic as a product of two linear factors.
When solving or factoring a quadratic equation, remember:

• The 'a' coefficient represents the term containing \(x^2\).
• The 'b' coefficient pertains to the linear \(x\) term.
• The 'c' coefficient is the constant term.

Understanding the structure of quadratic equations is crucial for processes like polynomial factoring.

Polynomial Factoring

Polynomial factoring is the process of rewriting a polynomial as a product of simpler polynomials. This technique simplifies expressions and solves equations by breaking them down into multiplicative factors. For the trinomial \(-15x^2 + 26x - 8\), the aim is to express it as a product of two binomial expressions.
The process involves several steps:

• Identify the structure of the polynomial and assign values to \(a\), \(b\), and \(c\).
• Find two numbers that multiply to \(a \times c\) and add up to \(b\).
• Split the middle term using these two numbers to create groups.
• Factor each group and find the common factor among them.

Through this method, complex polynomials are simplified into factors that are easier to manage.

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest number or expression that divides two or more numbers or expressions without leaving a remainder. In factoring polynomials, identifying and extracting the GCF is a crucial step to simplify expressions.
When factoring by grouping, each group often contains a GCF that needs to be extracted. This method is seen in the factorization process like in our trinomial example, \(-15x^2 + 20x\) and \(6x - 8\). By finding the GCF in each group, the expression is simplified.

• For \(-15x^2 + 20x\), the GCF is \(5x\).
• For \(6x - 8\), the GCF is \(2\).

Recognizing and factoring out the GCF from polynomial expressions is an important skill in algebra.

Binomial Expressions

Binomial expressions are polynomials with exactly two terms, such as \(3x - 4\). Factoring trinomials often results in identifying common binomial expressions within polynomial groups.
In the example with \(-15x^2 + 26x - 8\), after splitting and grouping terms, the common binomial \((3x - 4)\) emerges. Factoring out such binomials consolidates the polynomial into a simpler, more useful form.

• Identify patterns within the polynomial groups.
• Look for repeated terms or expressions.
• Extract these binomials as factors.

Mastering binomial expressions not only aids in polynomial factoring but also builds a foundation for more advanced algebraic concepts.