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Polynomials

Definition

A polynomial is an algebraic expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Examples: * 5x² - 3x + 7 (a trinomial) * 4y³ + 2y (a binomial) * -8 (a monomial, constant polynomial)

The degree of a polynomial is the highest exponent of its variable(s).

Prerequisites

To work with polynomials, you should have a strong foundation in:

  1. Variables & Expressions
  2. Integers & Real Numbers
  3. Exponents & Radicals (Exponent Rules)
  4. Order of Operations
  5. Combining Like Terms

Learning Objectives

After mastering this topic, you should be able to:

  1. Identify polynomials, their terms, coefficients, and degree.
  2. Classify polynomials by degree (constant, linear, quadratic, cubic, etc.) and number of terms (monomial, binomial, trinomial).
  3. Add and subtract polynomials by combining like terms.
  4. Multiply polynomials (monomial by polynomial, binomial by binomial using FOIL, polynomial by polynomial).
  5. Divide polynomials using long division and synthetic division (optional).
  6. Apply polynomial operations to solve problems.

Key Operations

Addition and Subtraction

  • Combine like terms (terms with the same variable and the same exponent).
  • Example: (3x² + 2x - 5) + (x² - 4x + 8)
    • = (3x² + x²) + (2x - 4x) + (-5 + 8)
    • = 4x² - 2x + 3

Multiplication

  • Monomial × Polynomial: Use the distributive property.
    • Example: 2x(x² + 3x - 1) = 2x(x²) + 2x(3x) + 2x(-1) = 2x³ + 6x² - 2x
  • Binomial × Binomial (FOIL): First, Outer, Inner, Last.
    • Example: (x + 2)(x + 5) = x(x) + x(5) + 2(x) + 2(5) = x² + 5x + 2x + 10 = x² + 7x + 10
  • Polynomial × Polynomial: Distribute each term of the first polynomial to every term of the second polynomial.
    • Example: (x + 3)(x² - x + 4)
      • = x(x² - x + 4) + 3(x² - x + 4)
      • = (x³ - x² + 4x) + (3x² - 3x + 12)
      • = x³ + (-x² + 3x²) + (4x - 3x) + 12
      • = x³ + 2x² + x + 12

Division (More Advanced)

  • Long Division: Similar process to numerical long division.
  • Synthetic Division: A shortcut method for dividing by a linear binomial of the form (x - c).

Examples

Example 1: Addition (Middle School/High School)

Problem: Subtract (2y² - y + 6) from (5y² + 3y - 1). Solution: (5y² + 3y - 1) - (2y² - y + 6) = 5y² + 3y - 1 - 2y² + y - 6 = (5y² - 2y²) + (3y + y) + (-1 - 6) = 3y² + 4y - 7

Example 2: Multiplication (FOIL) (High School)

Problem: Multiply (2x - 3)(x + 4). Solution: First: (2x)(x) = 2x² Outer: (2x)(4) = 8x Inner: (-3)(x) = -3x Last: (-3)(4) = -12 Combine: 2x² + 8x - 3x - 12 = 2x² + 5x - 12

Example 3: Polynomial Multiplication (High School)

Problem: Find the product: (a - 1)(a² + a + 1) Solution: a(a² + a + 1) - 1(a² + a + 1) = (a³ + a² + a) - (a² + a + 1) = a³ + a² + a - a² - a - 1 = a³ + (a² - a²) + (a - a) - 1 = a³ - 1 (This is a difference of cubes pattern)

Common Misconceptions

  1. Errors Combining Terms: Only combine terms with the exact same variable and exponent.
  2. Sign Errors: Especially when subtracting polynomials or multiplying negative terms.
  3. FOIL Errors: Forgetting one of the terms (often Outer or Inner) or combining incorrectly.
  4. Exponent Errors during Multiplication: Forgetting to add exponents when multiplying like bases (x² ⋅ x = x³).

Applications in SAT

Polynomial operations are fundamental in the SAT Math section:

  1. Algebraic Manipulation: Adding, subtracting, multiplying polynomials to simplify expressions.
  2. Solving Equations: Factoring polynomials (especially quadratics) to find roots.
  3. Function Analysis: Understanding the behavior of polynomial functions, finding intercepts, end behavior.
  4. Equivalency: Determining if two polynomial expressions are equivalent.

Advanced Connections

Polynomials are central to many areas of mathematics:

  1. Factoring - Breaking polynomials into simpler factors.
  2. Quadratic Equations - Polynomial equations of degree 2.
  3. Advanced Functions - Analyzing graphs and properties of higher-degree polynomial functions.
  4. Rational Expressions: Fractions involving polynomials.
  5. Calculus: Finding derivatives and integrals of polynomial functions.

Practice Problems

  1. Basic: Simplify: (4p³ - 2p² + 5) + (-p³ + 6p² - p)
  2. Intermediate: Multiply: (3x - 2)(2x + 5)
  3. Advanced: Multiply: (y - 4)(y² + 2y - 3)
  4. SAT-Level: If (ax + b)(cx - d) = 6x² - 5x - 4 for all values of x, where a, b, c, and d are constants, what is the value of ac + bd?