Operations with Polynomials: Foundations

[Video: Introduction to Polynomial Operations]

Learn how to add, subtract, multiply, and factor polynomials - essential skills for solving higher-level SAT math problems.

Polynomial Basics

What is a Polynomial?

A mathematical expression with one or more terms where variables have non-negative integer exponents.

Examples:

Monomial: 3x²
Binomial: x + 5
Trinomial: 2x² - 3x + 1

Polynomial Operations

Addition and Subtraction

Combine like terms (terms with the same variable and exponent).

Example: (3x² + 2x - 5) + (x² - 4x + 1)

= (3x² + x²) + (2x - 4x) + (-5 + 1)

= 4x² - 2x - 4

Multiplication

Use the distributive property (FOIL method for binomials).

Example 1: 3x(x² - 2x + 4)

= 3x³ - 6x² + 12x

Example 2: (x + 3)(x - 2)

= x(x) + x(-2) + 3(x) + 3(-2)

= x² - 2x + 3x - 6

= x² + x - 6

Special Products

Pattern	Formula
Difference of Squares	(a + b)(a - b) = a² - b²
Square of a Binomial	(a ± b)² = a² ± 2ab + b²

Example: (2x + 5)²

= (2x)² + 2(2x)(5) + 5²

= 4x² + 20x + 25

Factoring Polynomials

Greatest Common Factor (GCF)

Factor out the largest common term.

Example: 6x³ - 9x²

= 3x²(2x - 3)

Factoring Quadratics

For x² + bx + c, find two numbers that multiply to c and add to b.

Example: x² - 5x + 6

Numbers: -2 and -3 (since -2 × -3 = 6 and -2 + -3 = -5)

= (x - 2)(x - 3)

Key Takeaways

Combine only like terms when adding/subtracting
Use distributive property thoroughly when multiplying
Memorize special product formulas to save time
Always look for GCF first when factoring
Check your work by expanding factored forms

Practice Question

Which of the following is equivalent to (2x + 3)(x - 4) - (x² - 5x)?

A) x² - 6x - 12

B) x² + 6x - 12

C) x² - 10x - 12

D) x² + 10x - 12

Follow these steps:

First multiply (2x + 3)(x - 4) using FOIL
Then subtract (x² - 5x) from the result
Combine like terms to simplify

Step-by-Step Solution:

1. Multiply (2x + 3)(x - 4):

= 2x(x) + 2x(-4) + 3(x) + 3(-4)

= 2x² - 8x + 3x - 12

= 2x² - 5x - 12

2. Subtract (x² - 5x):

= (2x² - 5x - 12) - (x² - 5x)

= 2x² - 5x - 12 - x² + 5x

3. Combine like terms:

= (2x² - x²) + (-5x + 5x) - 12

= x² - 12

Wait a minute - this doesn't match any options! There seems to be an error in the question options. In a real SAT scenario, one of the options would be x² - 12.