Problem 17 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 10 x^{2}-9 x+2 $$

Step-by-Step Solution

Answer

The trinomial factors as \((5x - 2)(2x - 1)\).

Step 1: Identify the Trinomial Coefficients

First, we need to identify the coefficients in the trinomial given as \(10x^2 - 9x + 2\). Here, the coefficient \(a\) of \(x^2\) is 10, \(b\) of \(x\) is -9, and the constant term \(c\) is 2.

Step 2: Multiply a and c

Next, multiply the coefficients \(a\) and \(c\). That is \(10 \times 2 = 20\). This product will help us find two numbers that add to \(b\).

Step 3: Find Two Numbers that Multiply to 20 and Add to -9

Look for two numbers that multiply to 20 (the result from Step 2) and add to -9 (coefficient \(b\)). The two numbers that satisfy these conditions are -5 and -4 since \(-5 \times -4 = 20\) and \(-5 + -4 = -9\).

Step 4: Rewrite the Middle Term

Rewrite the trinomial by breaking the middle term using the two numbers found in Step 3. So, \(10x^2 - 9x + 2\) becomes \(10x^2 - 5x - 4x + 2\).

Step 5: Group the Terms

Group the terms in pairs: \((10x^2 - 5x) + (-4x + 2)\).

Step 6: Factor by Grouping

Factor out the greatest common factor from each group. In the first group \((10x^2 - 5x)\), factor out \(5x\), giving \(5x(2x - 1)\). In the second group \((-4x + 2)\), factor out \(-2\), giving \(-2(2x - 1)\).

Step 7: Combine the Factored Form

Now, since both terms share a common factor \((2x - 1)\), combine them: \((5x - 2)(2x - 1)\). This is the factored form of the trinomial.

Key Concepts

Polynomial Coefficients

In algebra, a polynomial is an expression made up of variables, coefficients, and exponents, all combined using mathematical operations like addition, subtraction, and multiplication. The coefficients in a polynomial are the numerical factors in terms of the expression. For instance, in the trinomial \(10x^2 - 9x + 2\), we have three coefficients:

• \(a = 10\), which is the coefficient of \(x^2\)
• \(b = -9\), which is the coefficient of \(x\)
• \(c = 2\), which is the constant term

Understanding these coefficients is crucial, as they dictate the polynomial's characteristics and behavior.
In solving problems like factoring a trinomial, correctly identifying coefficients \(a\), \(b\), and \(c\) is a vital first step.
This initial identification helps us organize the subsequent steps, ensuring a systematic approach to finding a solution.

Factor by Grouping

Factor by grouping is a useful method for factoring certain types of algebraic expressions, particularly trinomials. The key to this technique is to reorganize the expression so it can be split into groups that can be easily factored.
Here's a simple breakdown of the process:

• First, identify a common factor from groups of terms that you've chosen to pair up.
• Factor those groups separately by pulling out the greatest common factor from each.
• Finally, look for a common binomial factor between the pairs and factor it out.

For instance, in the trinomial \(10x^2 - 9x + 2\), once rewritten as \((10x^2 - 5x) + (-4x + 2)\), the terms in each pair can be factored individually:

• From \(10x^2 - 5x\), factor out \(5x\) to get \(5x(2x - 1)\).
• From \(-4x + 2\), factor out \(-2\) to get \(-2(2x - 1)\).

Combining these gives \((5x - 2)(2x - 1)\), and factor by grouping is successfully completed.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators like addition, subtraction, multiplication, and division. These forms are foundational in algebra and are vital for solving equations and understanding mathematical relationships.
In an expression like \(10x^2 - 9x + 2\), "\(x\)" represents a variable, a placeholder for values that we can plug into the expression.
The challenge often lies in finding ways to simplify or alter these expressions, such as factoring them, to unlock solutions or insights.

• Factoring changes the appearance of an algebraic expression, which can reveal properties or solutions that weren’t obvious before.
• Sometimes, expressions that seem complex can be broken down into simpler parts using methods like factoring.

Understanding how to manipulate algebraic expressions is one of the major skills you’ll need in algebra.

Quadratic Equations

Quadratic equations represent a type of polynomial equation where the highest degree of the variable is squared. They take the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
These equations can be solved through various methods, including factoring, which is often the simplest and most direct when applicable.
Solving via factoring involves rewriting the equation as a product of two binomials, such as \((5x - 2)(2x - 1) = 0\).
Once factored, each binomial is set to zero and solved for \(x\), the variable.

• This yields potential solutions to the original equation, which are often called the equation's roots or zeros.
• Solving quadratic equations by factoring leverages the zero product property, where if the product of two numbers is zero, at least one of the numbers must be zero.

Mastering quadratic equations is fundamental for progressing in algebra since they model many real-world problems and scenarios.