Problem 11 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. \(15 x^{2}-23 x+4\) a. Find two numbers whose product is \(15 \cdot 4=60\) and whose sum is -23 . b. Write \(-23 x\) using the factors from part (a). c. Factor by grouping.

Step-by-Step Solution

Answer

The factored form is \((3x - 4)(5x - 1)\).

Step 1: Understand the Problem

The task is to factor the trinomial \(15x^2 - 23x + 4\) by grouping. We'll break it down into parts as guided.

Step 2: Calculate Product and Sum

First, we need to find two numbers whose product is \( 15 \times 4 = 60 \) and whose sum is \(-23\). This will help us break down the middle term.

Step 3: Find the Pair of Numbers

To satisfy both conditions, we look for two numbers: \(-20\) and \(-3\). These numbers multiply to \(60\) (since \(-20 \times -3 = 60\)) and add up to \(-23\) (since \(-20 + -3 = -23\)).

Step 4: Rewrite the Middle Term

Use the numbers found in Step 3 to decompose \(-23x\). Thus, rewrite the expression as: \(15x^2 - 20x - 3x + 4\).

Step 5: Factor by Grouping

Now, group the terms for factoring: \((15x^2 - 20x) - (3x - 4)\). Factor out the common factors in each group. In \(15x^2 - 20x\), the common factor is \(5x\), giving us \(5x(3x - 4)\). In \(-3x + 4\), factor out \(-1\), giving us \(-1(3x - 4)\).

Step 6: Combine and Factor the Expression

Notice \(3x - 4\) is common in both groups. Factor it out: \((3x - 4)(5x - 1)\).

Step 7: Final Verification

Multiply the factors \((3x - 4)\) and \((5x - 1)\) to ensure they give back the original polynomial \(15x^2 - 23x + 4\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that involve variables raised to different powers and combined using addition or subtraction. They play a fundamental role in algebra, providing a way to express relationships and solve various algebraic problems. A trinomial is a type of polynomial consisting of three terms.

• The standard form of a trinomial is: ax² + bx + c, where a, b, and c are constants.
• In our example, the trinomial expression is 15x² - 23x + 4.

When working with polynomials, the goal often involves simplifying or factoring them to reveal important properties or solutions. Understanding the structure of polynomial expressions helps in applying techniques like factoring.

Grouping Method

The grouping method is a useful factoring technique for polynomials, especially trinomials, where the goal is to break down and factor the expression into simpler components.
Here’s how the grouping method works in practice:

• First, identify two numbers that multiply to give the product of the first and last coefficients (in this case, 15 and 4) and add up to the middle coefficient.
• Rewrite the middle term using these two numbers to split it into two terms.

For the trinomial 15x² - 23x + 4:

• We looked for numbers that multiply to 60 (15 * 4) and sum to -23, which are -20 and -3.
• Rewriting the expression: 15x² - 20x - 3x + 4.
• By grouping terms, we created: (15x² - 20x) and (-3x + 4).

This method relies on visually verifying common factors in the separated groups.

Algebra Techniques

Algebra involves various techniques to manipulate expressions and solve equations, with factoring being one of the key methods. The art of factoring transforms a polynomial into a product of simpler expressions, aiding in solving equations or simplifying algebraic fractions.
Here’s an outline of important algebra techniques used in factoring by grouping:

• Identify the greatest common factor in each group of terms: In the expression 15x² - 20x, the greatest common factor is 5x.
• Similarly, in -3x + 4, factor out -1.
• Re-factor the entire expression by taking out the common binomial factor: In this example, 3x - 4 was common in both groups.

By employing these techniques, we factored the trinomial 15x² - 23x + 4 into its component binomials: (3x - 4)(5x - 1). Confirming that multiplying these binomials returns the original expression ensures that the factoring was performed correctly.