\begin{definition}[Polynomial] A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. \end{definition}
\begin{callout}[Note] The degree of a polynomial is the highest power of the variable in the expression. \end{callout}
Classifying Polynomials
Polynomials are classified based on the number of terms they contain:
- Monomial: A polynomial with one term.
- Binomial: A polynomial with two terms.
- Trinomial: A polynomial with three terms.
- Polynomial: A general term for expressions with one or more terms.
\begin{callout}[Example]
- \$3x^2\$ is a monomial.
- \$2x + 5\$ is a binomial.
- \$x^2 - 4x + 7\$ is a trinomial.
- \$4x^3 - 2x^2 + x - 5\$ is a polynomial. \end{callout}
Polynomial Arithmetic
Polynomials can be added, subtracted, multiplied, and divided. These operations follow specific rules to ensure the result is also a polynomial.
\begin{callout}[Note] The degree of a polynomial is the highest power of the variable in the expression. \end{callout}
Multiplying a Polynomial by a Constant
To multiply a polynomial by a constant, multiply each term of the polynomial by the constant.
\begin{callout}[Example] Multiply \$3x^2 - 2x + 5\$ by \$4\$:
- Multiply each term by \$4\$: \$4 \cdot (3x^2 - 2x + 5)\$.
- Result: \$12x^2 - 8x + 20\$. \end{callout}
Adding Polynomials
To add polynomials, combine like terms. Like terms have the same variable raised to the same power.
\begin{callout}[Example] Add \$(x^2 - 5x + 6)\$ and \$(2x^2 + 3x - 2)\$:
- Combine \$x^2\$ terms: \$x^2 + 2x^2 = 3x^2\$.
- Combine \$x\$ terms: \$-5x + 3x = -2x\$.
- Combine constants: \$6 - 2 = 4\$.
- Result: \$3x^2 - 2x + 4\$. \end{callout}
Subtracting Polynomials
Subtracting polynomials involves distributing the negative sign and then combining like terms.
\begin{callout}[Example] Subtract \$(x^2 - 2x + 3)\$ from \$(3x^2 + 5x - 1)\$:
- Distribute the negative sign: \$3x^2 + 5x - 1 - (x^2 - 2x + 3)\$ becomes \$3x^2 + 5x - 1 - x^2 + 2x - 3\$.
- Combine like terms: \$2x^2 + 7x - 4\$. \end{callout}
\begin{callout}[CommonMistake] Forgetting to distribute the negative sign when subtracting polynomials is a common error. \end{callout}
Multiplying Polynomials
Multiplying polynomials involves multiplying each term in one polynomial by each term in the other polynomial, then combining like terms.
\begin{callout}[Example] Multiply \$(x + 3)(2x^2 - 4x + 5)\$:
- Multiply \$x\$ by each term in \$2x^2 - 4x + 5\$: \$x \cdot 2x^2 = 2x^3\$, \$x \cdot (-4x) = -4x^2\$, \$x \cdot 5 = 5x\$.
- Multiply \$3\$ by each term in \$2x^2 - 4x + 5\$: \$3 \cdot 2x^2 = 6x^2\$, \$3 \cdot (-4x) = -12x\$, \$3 \cdot 5 = 15\$.
- Combine all terms: \$2x^3 - 4x^2 + 5x + 6x^2 - 12x + 15\$.
- Result: \$2x^3 + 2x^2 - 7x + 15\$. \end{callout}
\begin{callout}[Tip] When multiplying polynomials, ensure every term in the first polynomial is multiplied by every term in the second polynomial. \end{callout}
Multiplication Patterns
There are special patterns for multiplying certain types of polynomials, such as perfect squares and difference of squares.
Perfect Square Pattern
For a binomial \$(x + a)^2\$, the expansion is \$x^2 + 2ax + a^2\$.
\begin{callout}[Example] Simplify \$(x - 3)^2\$:
- Use the pattern: \$x^2 + 2(-3)x + (-3)^2\$.
- Result: \$x^2 - 6x + 9\$. \end{callout}
Difference of Squares Pattern
For binomials \$(x - a)(x + a)\$, the expansion is \$x^2 - a^2\$.
\begin{callout}[Example] Simplify \$(x - 3)(x + 3)\$:
- Use the pattern: \$x^2 - 3^2\$.
- Result: \$x^2 - 9\$. \end{callout}
\begin{callout}[Self_Review]
- Classify the polynomial \$4x^3 - 2x + 7\$.
- Multiply \$(2x + 3)(x^2 - x + 4)\$ and simplify.
- Use the difference of squares pattern to simplify \$(x - 5)(x + 5)\$. \end{callout}
\begin{callout}[Tok] How do the patterns in polynomial multiplication reflect the underlying structure of algebra? Can these patterns be extended to higher dimensions or other mathematical systems? \end{callout}
