Factoring Algebraic Expressions

Factoring algebraic expressions is a fundamental concept in algebra that involves breaking down an expression into simpler terms called factors.

The process of factoring is akin to finding the prime factors of a number but applied to algebraic expressions. Factoring allows us to simplify expressions, solve equations, identify common factors, and understand the structure of polynomials.

In essence, when we factor an algebraic expression, we rewrite it as a product of simpler expressions. This can involve various techniques depending on the nature of the expression, such as factoring out common terms, using special products and identities, or employing specific methods like grouping.

Factoring out the Greatest Common Factor (GCF)

This involves identifying and factoring out the largest common factor from all the terms in the expression.

Example

Factor the the expression 6x^2 + 9x

Solution

GCF is 3x. So, we can factor out to get 3x(2x + 3)

Factoring Special Products

Special products like the difference of squares (a^2 – b^2), perfect square trinomials (a^2 + 2ab + b^2 or a^2 – 2ab + b^2), and the sum/difference of cubes can be factored using specific formulas or patterns.

Example

Expression 4x^2 – 9y^2 can be factored as

Solution

4x^2 – 9y^2 = (2x)^2 – (3y)^2 = (2x + 3y)(2x – 3y)

Factoring Trinomials

Trinomials are expressions with three terms. Factoring them usually involves finding two binomials that multiply together to give the original trinomial. Techniques like trial and error, grouping, or special methods like the AC method can be used.

Example

Factor the trinomial x^2 + 5x + 6

Solution

We need two numbers that multiply to give us 6 and add up to 5. These numbers are 2x and 3x. So the middle term can be split as x^2 + 2x + 3x + 6

Grouping the terms, we get (x^2 + 2x) + (3x + 6)

Factoring out GCF from each group, we get x(x + 2) + 3(x + 2)

Taking common (x + 2) outside, we have (x + 2)(x + 3)

Factoring by Grouping

In some expressions with four or more terms, grouping pairs of terms together and factoring out common factors from each pair can simplify the expression, eventually leading to a common factor that can be factored out further.

Example

Factor the expression 2x^3 + 4x^2 + 3x + 6

Solution

Grouping the terms, we get (2x^3 + 4x^2) + (3x + 6)

Factoring out GCF from each group, we get 2x^2(x + 2) + 3(x + 2)

Taking common (x + 2) outside, we have (x + 2)(2x^2 + 3)

Factoring Quadratic Trinomials

Quadratic trinomials are trinomials with a leading coefficient of 1. These can often be factored using techniques like trial and error, factoring by grouping, or specific methods like completing the square or using the quadratic formula.

The quadratic formula that can be used to find the factors is :
x = [-b ± √(b^2 – 4ac)]/2a

Example

Factor the expression x^2 – 8x + 12

Solution

Equate the expression to zero and find the solutions of x first.

The roots of the equation x^2 – 8x + 12 = 0 by using the above formula is 6 or 2. So, the factors of the expression x^2 – 8x + 12 is (x – 6)(x – 2)

Factoring Algebraic Expressions FAQS

Grade 7 Science Worksheets

• Elements and Compounds
• Solar Energy
• Photosynthesis
• Electricity and Magnetism
• Law of conservation of energy
• Periodic table
• Properties of Matter
• Waves
• Energy Resources 
• Weather and Climate
• Immune, Circulatory and Digestive Systems
• Organs in Multi-cellular Organism
• Sedimentary, Igneous, and Metamorphic Rocks
• Structure of the Earth
• Law of Conservation of Mass
• Physical and Chemical Changes
• Scientific Method
• Human Digestive System
• Environmental Science
• Renewable and Non-renewable energy Resources
• Characteristics of Living Organisms
• Life Science
• Earth and Space Science
• Solar Eclipse
• Heat Technology
• Newton’s Laws of Motions
• Physical Science
• Tools, Measurement and SI Units
• Earth Atmosphere
• Interactions of Living things
• The Earth Ecosystem
• Organelles in Plant and Animal cells
• Layers of the Earth
• Cycles in Nature

Grade 7 Math Worksheets

• Fractions
• Linear equations word problems
• Statistics
• Properties of Parallel Line
• Finding slope from an equation
• Identifying Quadrilaterals
• Percent Change
• Properties of addition and multiplication
• Pythagorean Theorem
• Solving two step inequalities
• Symmetry
• Fractions to Decimals (New)
• Whole Number Exponents with Integer Bases (New)
• Adding and Subtracting Fractions (New)
• Integer Addition and Subtraction (New)
• Dividing Mixed Numbers (New)
• Basics of Coordinate Geometry (New)