Polynomials are like building blocks in algebra. They're expressions with variables and exponents that we can add, subtract, and manipulate. Understanding how to work with polynomials is crucial for solving more complex math problems.

In this section, we'll cover the basics of polynomials, including their degree and how to find their values. We'll also learn how to add and subtract polynomials, which is essential for simplifying expressions and solving equations.

Polynomial Basics

Degree of polynomials

• Highest exponent of the variable determines the degree of a polynomial
  • $3x^2 + 2x - 1$ has a degree of 2 (highest exponent of $x$ is 2)
• For polynomials with multiple variables, degree is the sum of exponents in each term
  • $2x^2y + 3xy^2 - 4$ has a degree of 3 ($2x^2y$ has exponents 2 and 1, summing to 3)
• Constant polynomials without variables have a degree of 0 ($5$, $-2$)

Value of polynomial functions

• To find the value of a polynomial function $f(x)$ for an input $a$, substitute $a$ for $x$ and simplify
  • If $f(x) = 2x^2 - 3x + 1$, to find $f(2)$:
    1. Replace $x$ with 2: $f(2) = 2(2)^2 - 3(2) + 1$
    2. Simplify: $f(2) = 2(4) - 6 + 1 = 8 - 6 + 1 = 3$
• Be careful with signs when substituting negative values
  • If $f(x) = x^2 - 4x + 3$, to find $f(-2)$:
    1. Replace $x$ with -2: $f(-2) = (-2)^2 - 4(-2) + 3$
    2. Simplify: $f(-2) = 4 + 8 + 3 = 15$

Polynomial Operations

Addition vs subtraction of polynomials

• Adding polynomials involves combining like terms (same variables and exponents)
  • $(2x^2 + 3x - 1) + (x^2 - 4x + 2)$ $= (2x^2 + x^2) + (3x - 4x) + (-1 + 2)$ $= 3x^2 - x + 1$
• Subtracting polynomials requires distributing the negative sign to each term in the subtracted polynomial, then combining like terms
  • $(5x^2 + x - 1) - (4x^2 - 2x + 3)$ $= (5x^2 + x - 1) + (-4x^2 + 2x - 3)$ $= (5x^2 - 4x^2) + (x + 2x) + (-1 - 3)$ $= x^2 + 3x - 4$
• Polynomials are examples of algebraic expressions, which are combinations of variables, constants, and operations

Combining polynomial functions

• To combine polynomial functions using addition or subtraction, perform the operation on the functions as if they were polynomials
  • If $f(x) = 3x^2 - 2x + 1$ and $g(x) = 2x^2 + 4x - 3$, then:
    • $(f + g)(x) = f(x) + g(x) = (3x^2 - 2x + 1) + (2x^2 + 4x - 3) = 5x^2 + 2x - 2$
    • $(f - g)(x) = f(x) - g(x) = (3x^2 - 2x + 1) - (2x^2 + 4x - 3) = x^2 - 6x + 4$
• The result is a new polynomial function ($5x^2 + 2x - 2$, $x^2 - 6x + 4$)

Properties of Polynomial Addition and Subtraction

• Commutative property: The order of addition doesn't affect the result (e.g., $a + b = b + a$)
• Associative property: Grouping of terms doesn't affect the result (e.g., $(a + b) + c = a + (b + c)$)
• These properties apply to the addition of polynomials and their terms