Problem 9 - Algebra and Trigonometry

Question

9–16 ? Factor the trinomial. $$ x^{2}+2 x-3 $$

Step-by-Step Solution

Answer

The factored trinomial is \((x + 3)(x - 1)\).

Step 1: Identify the Form

This problem involves a trinomial factorization, and the given expression is in the form of a quadratic trinomial: \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 2 \), and \( c = -3 \).

Step 2: Find Two Numbers

We need to find two numbers that multiply to \( a \times c = 1 \times (-3) = -3 \) and add up to \( b = 2 \). These numbers are \( 3 \) and \( -1 \) because \( 3 imes (-1) = -3 \) and \( 3 + (-1) = 2 \).

Step 3: Rewrite the Middle Term

Rewrite the expression by splitting the middle term using the numbers found in Step 2: \( x^2 + 3x - x - 3 \).

Step 4: Factor by Grouping

Group the terms: \( (x^2 + 3x) + (-x - 3) \). Factor out the greatest common factor from each group: \( x(x + 3) - 1(x + 3) \).

Step 5: Factor the Common Binomial

The expression \( x(x + 3) - 1(x + 3) \) shows a common binomial \( (x + 3) \). Therefore, factor it out: \( (x + 3)(x - 1) \).

Step 6: Verify the Factored Form

To ensure the factorization is correct, multiply \( (x + 3)(x - 1) \) to check if it returns to the original expression:\[(x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3\]The factorization is verified.

Key Concepts

Factoring by Grouping

Factoring by grouping is a technique used to factor certain algebraic expressions by grouping terms in a way that allows you to factor out common factors. This is particularly useful in quadratic expressions where direct factoring might not be easy. The aim is to break down the expression into smaller groups so that factoring becomes manageable.

Here's a step-by-step approach to factoring by grouping:

• First, identify pairs or groups within the expression that can be easily factored.
• Then, factor out any common factors from each group.
• Finally, ensure that each group has a common binomial factor that can be factored out.

In a quadratic trinomial like the one given, you might first split the middle term to form two groups and then utilize common factors to simplify into a factored form.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents a variable. The characteristic feature of a quadratic expression is that the highest power of the variable \( x \) is two.

To factor a quadratic expression effectively, it's essential to understand its structure:

• The coefficient \( a \) affects the disparity in the width of the parabola when drawn as a graph.
• The coefficient \( b \) controls the direction and position of the symmetry line.
• The constant \( c \) shifts the graph vertically.

For factoring purposes, such as in the problem given, you typically look for two numbers that multiply to \( ac \) and add to \( b \). Identifying these numbers allows you to recast the quadratic into a form that's simple to factor.

Algebraic Factorization

Algebraic factorization involves expressing an algebraic expression as a product of its factors. This skill is crucial for simplifying expressions, solving equations, and understanding functions deeper. In algebra, especially with quadratics, factorization simplifies complex structures to easily interpretable forms.

The process typically involves:

• Identifying the type of polynomial or expression you are dealing with.
• Looking for common factors or patterns, such as a difference of squares or perfect square trinomials.
• Applying methods like factoring by grouping to break down the expression into more reducible components.

In our quadratic expression \( x^2 + 2x - 3 \), algebraic factorization reduced it to \((x + 3)(x - 1)\). Doing so makes solving equations or analyzing properties much easier, highlighting the utility of mastering algebraic factorization.