Lesson 3: Polynomials and Long Division

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A polynomial of degree $n$ is a function of the form $$ P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{1}x + a_{0}$$ where $n$ is a positive integer and $a_{n} \neq 0$. The constants $a_{0}, a_{1}, \ldots, a_{n}$ are called coefficients. When the degree of $P(x)$ is $2$, the polynomial function is called a quadratic function. When the degree of $P(x)$ is $1$, the polynomial function is called a linear function.

Polynomial Arithmetic

When working with polynomial expressions, we can use the following:

• Addition/Subtraction: add/subtract coefficients of equal powers
• Multiplication: use the distributive property
• Division: use the long division algorithm, or synthetic division.

Powers of Polynomials

To quickly find the power of a polynomial we can use the binomial theorem. The first three powers of the binomial $(x+a)$ are given below.

• $(x+a)^{2} = x^{2}+2ax + a^{2}$
• $(x+a)^{3} = x^{3}+3ax^{2} + 3a^{2}x + a^{3}$
• $(x+a)^{4} = x^{4} + 4ax^{3} + 6a^{2}x^{2} + 4a^{3}x + a^{4}$