Problem 10 - Introductory Algebra for College Students

Question

Before getting to multiple-step factorizations, let's be sure that you are comfortable with exercises requiring only one of the factoring techniques. Factor each polynomial. $$3 x^{2}+2 x-5$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(3x^2 + 2x - 5\) is \((3x + 5)(x - 1)\)

Step 1: Identify the Coefficients and Constant

The quadratic polynomial is \(3x^2 + 2x - 5\). From this, the coefficients and constant are identified as: \(a = 3\), \(b = 2\), and \(c = -5\)

Step 2: Find Two Numbers

Find two numbers that multiply to -15 (the product of 3 and -5) and add up to 2. Those numbers are 5 and -3. This is because \(5 * -3 = -15\) and \(5 + -3 = 2\).

Step 3: Rewrite the Polynomial

The polynomial \(3x^2 + 2x - 5\) can be rewritten as \(3x^2 + 5x - 3x - 5\)

Step 4: Factor by Grouping

This step involves grouping and factoring out the greatest common factor from each group. The polynomial would then become \(x(3x + 5) - 1(3x + 5)\)

Step 5: Factor Out Common Binomial

The common binomial factor in the above step is \((3x + 5)\). Factor this out to get: \((3x + 5)(x - 1)\)

Key Concepts

Quadratic Polynomial

Quadratic polynomials are a specific type of polynomial equation that have the form \(ax^2 + bx + c\). The highest degree, which is the term with the highest power of the variable \(x\), is the squared term \(x^2\). This form of polynomial is particularly important in mathematics because it models a wide array of natural phenomena and geometric problems. Quadratic polynomials can have up to two real solutions, which are the values of \(x\) that make the equation equal zero. These solutions can be found by factoring, using the quadratic formula, or completing the square. Understanding how to manipulate quadratic polynomials is crucial for solving these equations and is a foundational element of algebra.

Factoring Techniques

Factoring techniques are essential tools used to simplify polynomials, particularly when solving quadratic equations. The main idea is to express the polynomial as a product of simpler terms, which can make solving the equation more straightforward and intuitive. Some common factoring techniques include:

• Factoring out the Greatest Common Factor (GCF): This involves identifying the largest polynomial that divides each term in the polynomial and factoring it out.
• Difference of Squares: A special case where a polynomial can be factored into two binomials if it can be expressed as the difference between two perfect squares.
• Trinomial Factoring: Often used for quadratic polynomials, this technique involves finding two numbers that multiply to \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(b\) (the linear coefficient).

Using these techniques efficiently requires practice and sometimes a bit of creativity in seeing the possible factors that can emerge from an expression.

Factoring by Grouping

Factoring by grouping is a clever technique often used for factoring polynomials that do not easily fit into simpler forms like trinomials or difference of squares. This method involves rearranging the polynomial into groups that each share a common factor, making it easier to factor them out. Let's break down the steps:

• Identify parts of the polynomial: Begin with separating the polynomial into distinct groups, often two pairs, as was done with the quadratic \(3x^2 + 5x - 3x - 5\).
• Factor each group: For each group, factor out the greatest common factor. Continuing with our example, this would be \(x(3x + 5)\) and \(-1(3x + 5)\).
• Extract the common factor: If a common binomial factor appears in both groups, factor it out, resulting in a simplified product of two factors, showcased as \((3x + 5)(x - 1)\).

This technique, while sometimes complex at first glance, becomes a powerful tool for handling polynomials that resist simpler methods of factoring.