Problem 20 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 2 x^{2}-7 x+3 $$

Step-by-Step Solution

Answer

The factored form is \((x - 3)(2x - 1)\).

Step 1: Identify the Terms

The given trinomial is \(2x^2 - 7x + 3\). The terms are: \(2x^2\), \(-7x\), and \(+3\).

Step 2: Find Two Numbers That Multiply to the Product of a and c

For the trinomial \(ax^2 + bx + c\), we need two numbers that multiply to \(a \cdot c = 2 \cdot 3 = 6\) and add up to \(b = -7\). These numbers are \(-1\) and \(-6\) because \((-1) \times (-6) = 6\) and \((-1) + (-6) = -7\).

Step 3: Rewrite the Middle Term Using the Two Numbers

Rewrite \(-7x\) as \(-1x - 6x\). This gives us:\[2x^2 - 1x - 6x + 3\].

Step 4: Group the Terms

Group the terms into two pairs:\[(2x^2 - 1x) + (-6x + 3)\].

Step 5: Factor by Grouping

Factor out the greatest common factor (GCF) from each group:- From \(2x^2 - 1x\), factor out \(x\): \(x(2x - 1)\).- From \(-6x + 3\), factor out \(-3\): \(-3(2x - 1)\).

Step 6: Combine the Factors

Now, since both terms contain the common factor \((2x - 1)\), factor it out:\[(x - 3)(2x - 1)\].

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like plus and minus). It's like a math sentence that defines a value or a set of values. Think of it as a recipe for calculating something. For example, the expression \(2x^2 - 7x + 3\) tells us how to combine the variable \(x\) with some numbers to get a result. When dealing with algebraic expressions, it's important to understand each part. In our trinomial \(2x^2 - 7x + 3\):

• \(2x^2\) is the quadratic term, where \(x\) is squared and multiplied by 2.
• \(-7x\) is the linear term, where \(x\) is not squared but multiplied by -7.
• \(+3\) is the constant term, independent of \(x\).

Learning to break down these expressions into parts helps simplify and solve them.

Polynomial Factoring

Polynomial factoring is an essential skill in algebra, especially when simplifying or solving expressions. It involves breaking down a complex polynomial into simpler, multiplied components. Imagine a polynomial as a big piece of furniture; factoring is like breaking it down into easy-to-handle parts.The goal in factoring polynomials is to find expressions that multiply together to form the original one. For example, in the trinomial \(2x^2 - 7x + 3\), we aim to express it as the product of simpler binomials, such as \((x - 3)(2x - 1)\).A well-factored polynomial can be very helpful in solving equations, especially quadratic ones, where we look for values that make the expression equal to zero. Just like solving a puzzle, polynomial factoring can make complicated problems more manageable.

Grouping Method

The grouping method is one strategy for factoring polynomials, particularly useful for quadratics. It involves organizing terms into manageable groups, making it easier to identify and pull out common factors.Let's walk through the process using \(2x^2 - 7x + 3\) as an example:

• Identify Pairs: Break the middle term into two parts to create pairs. Here, we rewrite \(-7x\) as \(-1x - 6x\).
• Grouping: Group terms by enclosing them. We get two pairs: \((2x^2 - 1x)\) and \((-6x + 3)\).
• Factoring Out Common Factors: For each group, find the greatest common factor. For \((2x^2 - 1x)\), it is \(x\), resulting in \(x(2x - 1)\). For \((-6x + 3)\), it is \(-3\), which gives us \(-3(2x - 1)\).
• Combine: Both groups include \((2x - 1)\), indicating it's common. So, factor this out, obtaining \((x - 3)(2x - 1)\).

This method is a systematic way of simplifying expressions like magic, turning complexity into clarity!

Quadratic Equations

Quadratic equations are mathematical equations that include a term with a variable raised to the power of two, usually expressed as \(ax^2 + bx + c = 0\). These equations often pop up in various math problems and real-world applications.Solving quadratic equations can involve:

• Factoring: Like we've seen, where an equation gets simplified into the product of binomials.
• Quadratic Formula: A formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) that directly provides the solution.
• Completing the Square: A method that reworks the equation to make it easier to solve.

Quadratic equations have a standard form and can be transformed to reveal their roots or solutions. These roots tell us where the graph of the equation crosses the x-axis, which is crucial for understanding its behavior. Learning these methods provides excellent tools for tackling various math challenges!