9th grade math - Algebra

Polynomials and Factoring

Polynomials and factoring are foundational topics in Algebra I and Algebra II. They help students understand the structure of algebraic expressions and how to simplify and solve complex equations. If you’ve been following our Algebra series at StudyMath.org, this article builds on your understanding of exponents and expressions covered previously.

What Is a Polynomial?

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents that are combined using addition, subtraction, and multiplication.

Example of a polynomial:

3x2 + 5x - 7

This expression has:

  • Three terms: 3x2, 5x, and -7
  • A degree of 2 (the highest exponent of x)

Terminology:

  • Term: Each part separated by + or –
  • Coefficient: The number in front of a variable (e.g., 3 in 3x2)
  • Degree: The highest exponent of the variable

Types of Polynomials

  • Monomial: One term (e.g., 5x)
  • Binomial: Two terms (e.g., x + 3)
  • Trinomial: Three terms (e.g., x2 + 2x + 1)

Understanding the type helps when selecting factoring strategies.

Factoring: The Reverse of Expanding

Factoring is the process of breaking a polynomial down into simpler “building blocks” called factors that, when multiplied together, result in the original polynomial.

For example:

x2 + 5x + 6 = (x + 2)(x + 3)

This shows that the trinomial can be written as a product of two binomials.

Common Factoring Techniques

1. Factoring Out the GCF (Greatest Common Factor)

Always check for the GCF first. Example:

6x2 + 9x = 3x(2x + 3)

2. Factoring Trinomials

To factor a trinomial like x2 + 5x + 6, find two numbers that multiply to 6 and add to 5 (2 and 3).

x2 + 5x + 6 = (x + 2)(x + 3)

3. Factoring Using Special Formulas

  • Difference of squares: a2 - b2 = (a - b)(a + b)
  • Perfect square trinomials: a2 + 2ab + b2 = (a + b)2

4. Factoring by Grouping

Used for 4-term polynomials:

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

Applications of Factoring

Factoring is essential for:

  • Solving quadratic equations: (x + 2)(x - 5) = 0 → x = -2 or 5
  • Simplifying rational expressions
  • Graphing parabolas and finding their zeros

Practice Problems

  1. Factor: x2 + 7x + 10
  2. Factor using GCF: 4x2 + 8x
  3. Factor completely: x2 - 16
  4. Group and factor: x3 + 3x2 + 2x + 6

Summary

Polynomials and factoring are core algebra skills that support solving equations, simplifying expressions, and analyzing functions. As you move into Algebra II, this foundation becomes even more critical for understanding complex math concepts.

Review previous articles on Exponents and continue exploring polynomial expressions on StudyMath.org.