🎠Introduction to Polynomials
In our earlier chapters, we looked at simple relationships like $y = kx$. A Polynomial is simply a more complex version of these algebraic expressions.
The word comes from:
- Poly- (meaning many)
- -nomial (meaning terms)
1. What is a Polynomial?
A polynomial is an expression consisting of variables (like $x$), coefficients (numbers in front of $x$), and exponents (powers).
The Golden Rules:
To be a polynomial, an expression CANNOT have:
- Negative or fractional exponents (No $x^{-2}$ or $x^{1/2}$).
- Variables in the denominator (No $1/x$).
- Variables under a square root (No $\sqrt{x}$).
2. Anatomy of a Polynomial
Let's look at $5x^3 - 2x + 7$:
- Terms: There are three terms ($5x^3$, $-2x$, and $7$).
- Degree: The highest exponent. Here, the degree is 3.
- Leading Coefficient: The number attached to the highest power ($5$).
- Constant: The number without a variable ($7$).
3. Classifying Polynomials
We name polynomials based on how many terms they have and their degree.
By Number of Terms:
- Monomial: 1 term (e.g., $4x^2$)
- Binomial: 2 terms (e.g., $3x + 5$)
- Trinomial: 3 terms (e.g., $x^2 - 4x + 4$)
By Degree:
| Degree | Name | Example | Graph Shape |
|---|---|---|---|
| 0 | Constant | $7$ | Horizontal Line |
| 1 | Linear | $2x + 3$ | Straight Line |
| 2 | Quadratic | $x^2 - 5$ | Parabola (U-shape) |
| 3 | Cubic | $x^3$ | S-curve |
4. Polynomials in Standard Form
Standard form means writing the terms in order from the highest exponent to the lowest.
- Messy: $5 + 2x^3 - x$
- Standard Form: $2x^3 - x + 5$
5. Why do we care?
In the real world, polynomials model everything from the trajectory of a basketball (Quadratic) to the growth of a savings account with compound interest.
Connection to Proportion: A linear polynomial with no constant ($y = mx$) is exactly what we studied in Direct Proportion! As the degree increases, the relationships just become more "curvy."