Factoring and Applications: Study Guide for Intermediate Algebra

Chapter 5: Factoring and Applications

5.1 The Greatest Common Factor; Factoring by Grouping

This section introduces the concept of factoring, focusing on finding the greatest common factor (GCF) and factoring by grouping. Factoring is a fundamental skill in algebra used to simplify expressions and solve equations.

• Definition: To factor a number means to break it down into multiplication of its prime factors.

• Prime Number: A number that has only two factors: 1 and itself.

• Factoring: The process of writing a number or expression as a product of its factors.

Finding the Greatest Common Factor (GCF):

1. List all factors of each term.

2. Identify the common factors.

3. The GCF is the largest factor common to all terms.

Example: Find the GCF of 12 and 18. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 GCF = 6

Factoring by Grouping: Used when there are four terms in an expression. Group terms in pairs and factor out the GCF from each pair.

• Step 1: Group terms in pairs.

• Step 2: Factor the GCF from each group.

• Step 3: If the resulting binomials are the same, factor them out.

Example: $x^2 + 2x + x + 2$ Group: $(x^2 + 2x) + (x + 2)$ Factor: $x(x + 2) + 1(x + 2)$ Final: $(x + 2)(x + 1)$

5.2 Factoring Trinomials

This section covers factoring trinomials of the form $ax^2 + bx + c$. The process involves finding two numbers that multiply to $ac$ and add to $b$.

• Trinomial: A polynomial with three terms.

• Factoring: Reverse the process of multiplying binomials ("unFOIL").

Example: Factor $x^2 + 5x + 6$ Find two numbers that multiply to 6 and add to 5: 2 and 3. Factor: $(x + 2)(x + 3)$

5.3 More on Factoring Trinomials

Trinomials with a leading coefficient other than 1 require special techniques. Two methods are commonly used: trial and error, and factor by grouping.

• Trial and Error: List possible pairs of factors and test combinations.

• Factor and Group: Split the middle term and group for factoring.

Example: Factor $3x^2 + 7x + 2$ Split: $3x^2 + x + 6x + 2$ Group: $(3x^2 + x) + (6x + 2)$ Factor: $x(3x + 1) + 2(3x + 1)$ Final: $(3x + 1)(x + 2)$

5.4 Special Factoring Techniques

Special techniques are used for factoring perfect squares, differences of squares, and differences of cubes.

• Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a + b)^2$

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

• Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Example: Factor $x^2 - 16$ $x^2 - 16 = (x + 4)(x - 4)$

Factoring Perfect Square Trinomials:

• Check if the first and last terms are perfect squares.

• Check if the middle term is twice the product of the square roots.

Example: $x^2 + 6x + 9 = (x + 3)^2$

5.5 Solving Quadratic Equations Using the Zero-Factor Property

Quadratic equations can be solved by factoring and applying the zero-factor property. If a product of factors equals zero, at least one factor must be zero.

• Zero-Factor Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Write the equation in standard form and factor.

• Set each factor equal to zero and solve for the variable.

Example: Solve $x^2 - 5x = 0$ Factor: $x(x - 5) = 0$ Solutions: $x = 0$, $x = 5$

5.6 Applications of Quadratic Equations

Quadratic equations are used to solve real-world problems, such as area, consecutive integers, and geometric applications.

• Assign a variable to the unknown.

• Write an equation based on the problem.

• Solve the equation by factoring.

• Check the solution in the context of the problem.

Example: The hypotenuse of a right triangle is 3 inches longer than the longer leg. The shorter leg is 3 inches shorter than the longer leg. Find the lengths of the sides.

Summary of Factoring Techniques

Factoring is a key skill in algebra, used to simplify expressions and solve equations. The main techniques include:

• Factoring out the GCF

• Factoring by grouping

• Factoring trinomials

• Special factoring formulas (difference of squares, cubes, perfect squares)

• Solving quadratic equations by factoring

Factoring Rule	Formula
Difference of Squares	$a^2 - b^2 = (a + b)(a - b)$
Perfect Square Trinomial	$a^2 + 2ab + b^2 = (a + b)^2$
Difference of Cubes	$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
Sum of Cubes	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Additional info: These notes cover all major factoring techniques and their applications, suitable for beginning-intermediate algebra students.