Problem 11 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. $$ 2 x^{2}-3 x-2 $$

Step-by-Step Solution

Answer

The factored form of the given trinomial \(2x^2 - 3x - 2\) is \((2x + 1)(x - 2)\).

Step 1: Analyze the Trinomial

The given trinomial is \(2x^2 - 3x - 2\). The coefficients of this trinomial are 2, -3, and -2, respectively.

Step 2: Find the Two Numbers

To find the two numbers, determine what two values multiplied together give you -4 (ac, where a is the coefficient of \(x^2\) and c is the constant term) and added together to give you -3 (b, the coefficient of x). The two numbers that satisfy these conditions are -4 and 1.

Step 3: Rewrite the Trinomial

Write the original trinomial as a four-term polynomial by breaking up the middle term using the two numbers found in Step 2: \(2x^2 - 4x + x - 2\).

Step 4: Factor by Grouping

Group the first two and last two terms and factor out the greatest common factor (GCF) from each group: \(2x(x - 2) + 1(x - 2)\).

Step 5: Factor out the Common Binomial

Since the quantity \(x - 2\) is common in both terms, it can be factored out: \((2x + 1)(x - 2)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like sentences in math with numbers and letters. They use symbols such as addition (+), subtraction (-), multiplication (×), and division (÷). In our trinomial, \(2x^2 - 3x - 2\), the numbers (or coefficients) and variables (x) are combined. An expression does not have an equals sign, unlike equations.
Recognizing different parts of an algebraic expression helps, such as:

• **Terms**: These are parts separated by plus or minus signs. Our equation has three terms: \(2x^2\), \(-3x\), and \(-2\).
• **Coefficients**: Numbers before the variables, like 2 in \(2x^2\). They indicate how many times to multiply the variable.
• **Constants**: Numbers without variables, like -2. They don’t change because they’re not tied to a variable.

Understanding these components aids in simplifying and factoring expressions, unlocking the power of algebra.

Quadratic Equations

Quadratic equations are essential in algebra and appear in the form \(ax^2 + bx + c = 0\). The equation links directly to its expression counterpart, like our trinomial \(2x^2 - 3x - 2\).
Quadratics are essential because they model various real-world scenarios, like projectile motions or areas. The standard quadratic equation includes:

• **Quadratic Term (\(ax^2\))**: Defined by the variable squared, tells us about the curve called a parabola.
• **Linear Term (\(bx\))**: Gives the line its slope, influencing direction.
• **Constant Term (\(c\))**: Moves the curve up or down within the coordinate plane.

The goal in many algebra problems, like ours, is to determine solutions (roots) where the equation equals zero. Here, factoring allows finding these roots quickly.

Polynomial Factorization

Polynomial factorization transforms complex expressions into simpler, multiplied parts. Think of breaking down a tough math problem into easy-to-handle pieces.
Our trinomial, \(2x^2 - 3x - 2\), can be factored into \((2x + 1)(x - 2)\). Here’s how factoring works:

• **Identify and separate** terms based on coefficients and constants.
• **Find pairs** that multiply to the product of the first and last coefficients, aiding in separating middle terms.
• **Regroup** and factor by grouping like parts together.
• **Common terms** are pulled out to reveal factored forms.

This process transforms the original polynomial into a product of factors, allowing us to solve for zero. It effectively simplifies equations and is crucial for solving quadratics. Further, mastering factorization provides tools for graphing, simplifying, and predicting outcomes in math and beyond.