Algebra - Addition, Subtraction, and Multiplication of Polynomials

Understanding Polynomials: A polynomial is an algebraic expression consisting of variables, coefficients, and exponents. Imagine a polynomial like $4x^3 - 2x + 7$ as a chain of distinct 'links' or terms separated by plus or minus signs. Each term has a degree, which is the value of its exponent.

Identifying Like Terms: Like terms are terms that have the exact same variable parts raised to the same powers, such as $5x^2$ and $-2x^2$. Visually, these can be thought of as items of the same 'category' or 'shape' that can be combined, whereas $x^2$ and $x$ are like different shapes (e.g., a square vs. a line) and cannot be added together.

Standard Form: Polynomials are typically written in descending order of the exponent's value. For example, $5x^3 + 2x^2 - x + 4$ is in standard form. Visually, the exponents create a 'staircase' effect going down from left to right, making the expression easier to read and calculate.

Addition of Polynomials: To add polynomials, you remove the parentheses and group like terms together. Imagine merging two separate sets of items and then sorting the identical items into single piles to simplify the total count.

Subtraction and the Negative Distributive Property: When subtracting a polynomial, you must distribute the negative sign to every term inside the parentheses. Visually, a minus sign before a bracket acts like a 'sign-flipper' or 'mirror' that reverses the sign of everything inside, turning $+( \dots )$ into $-( \dots )$ and $-( \dots )$ into $+( \dots )$.

The Distributive Property: This is the foundation of polynomial multiplication, where a term outside a bracket multiplies every term inside: $a(b + c) = ab + ac$. Visually, this is like a delivery person visiting every 'house' (term) inside the 'block' (parentheses).

Multiplying Binomials (FOIL): When multiplying two binomials, follow the FOIL method: Firsts, Outers, Inners, Lasts. Visually, you can draw four distinct arcs connecting the terms in the first bracket to those in the second to ensure every combination is multiplied.

Degree and Leading Coefficient: The degree is the highest exponent in the polynomial, and the leading coefficient is the number in front of that term. In a graph, the degree determines the maximum number of times the curve can cross the x-axis.