Polynomials: The Shape of an Equation

🧮2.1 What actually counts as a polynomial?

A polynomial is basically an algebraic expression built only from a variable raised to whole, non-negative powers (x⁰, x¹, x², x³, …), multiplied by numbers and added together. The degree of a polynomial p(x) is just the highest power of x that appears in it.

📛Naming polynomials by degree

• Linear polynomial — degree 1, general form ax + b (a ≠ 0). E.g. 2x − 3, √3x + 5.
• Quadratic polynomial — degree 2, general form ax² + bx + c (a ≠ 0). The name comes from "quadrate", meaning square.
• Cubic polynomial — degree 3, general form ax³ + bx² + cx + d (a ≠ 0).

🎰Plugging in a number: p(k)

For a linear polynomial ax + b, finding the zero is simple algebra: set ax + b = 0, which gives k = −b/a. So the zero of a linear polynomial is always −(constant term) ÷ (coefficient of x).

That raises a natural question this whole chapter answers: is a polynomial's zero always connected to its coefficients this neatly, even for quadratics and cubics? (Spoiler: yes — Section 2.3 below.)

📈2.2 What zeroes actually look like on a graph

Zeroes aren't just an algebra trick — they have a very visual meaning. If you plot y = p(x) on a graph, the zeroes of p(x) are exactly the points where the curve crosses (or touches) the x-axis.

📏Linear polynomials: always exactly one crossing

The graph of y = 2x + 3 is a straight line. Look at a couple of points on it:

This line crosses the x-axis at x = −3/2 — exactly matching the algebraic zero of 2x + 3 we'd get from −b/a = −3/2. In general, a straight (non-horizontal) line can only ever cross the x-axis at one point, so a linear polynomial always has exactly one zero.

🎢Quadratic polynomials: the parabola

Now plot y = x² − 3x − 4:

This U-shaped curve is called a parabola — it opens upward when a > 0, and downward like an upside-down U when a < 0. The curve crosses the x-axis exactly at x = −1 and x = 4, which are precisely the zeroes we found algebraically above.

🌊Cubic polynomials: up to 3 crossings

Cubic curves can wiggle more. Take y = x³ − 4x:

This curve crosses the x-axis three separate times, at −2, 0 and 2 — all three are zeroes of x³ − 4x. But not every cubic uses up all three "slots": y = x³ touches the x-axis at just x = 0 (a triple repeated zero, still only one crossing point), and y = x³ − x² crosses at only two points, 0 and 1.

🔗2.3 The secret code linking zeroes and coefficients

Here's the payoff of the whole chapter: once you know a polynomial's coefficients, you can predict facts about its zeroes without ever solving for them — and vice versa.

Quadratic case: worked example

Take p(x) = 2x² − 8x + 6. Splitting the middle term:

So the zeroes are 1 and 3. Now check the pattern:

This isn't a coincidence — it's always true. For any quadratic ax² + bx + c with zeroes α and β:

Example: x² + 7x + 10

Example: x² − 3 (irrational zeroes)

Using the identity a² − b² = (a−b)(a+b):

🏗️Building a quadratic backwards from sum and product

You can also run this in reverse: given a desired sum and product of zeroes, build the quadratic. Say sum = −3, product = 2:

Any polynomial of the form k(x² + 3x + 2), for any real k ≠ 0, also works — scaling the whole expression doesn't change where it crosses zero.

🎲Cubic case: three relationships, not two

For a cubic ax³ + bx² + cx + d with zeroes α, β, γ, there are now three relationships to track — including one easy to forget, the sum of products taken two at a time:

Example: 2x³ − 5x² − 14x + 8

Checking, this cubic's zeroes turn out to be 4, −2, and 1/2. Verifying all three relationships:

📋2.4 Summary

1. 📛 Polynomials of degree 1, 2 and 3 are called linear, quadratic, and cubic respectively.
2. 🧮 A quadratic polynomial has the form ax² + bx + c, with a, b, c real and a ≠ 0.
3. 📈 The zeroes of p(x) are exactly the x-coordinates where the graph of y = p(x) meets the x-axis.
4. 🎯 A quadratic has at most 2 zeroes; a cubic has at most 3 zeroes.
5. 🔗 For a quadratic with zeroes α, β: α + β = −b/a, and αβ = c/a.
6. 🎲 For a cubic with zeroes α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, and αβγ = −d/a.

TOOL 1 Polynomial Grapher & Zero Counter

Enter coefficients for a linear, quadratic, or cubic polynomial and see its graph plotted live, with every x-axis crossing (real zero) automatically marked and counted.

TOOL 2 Zeroes ↔ Coefficients Verifier

Enter the coefficients of a quadratic (a, b, c) or cubic (a, b, c, d) polynomial. The tool solves for the real zeroes and shows the sum/product (and, for cubics, sum-of-pairs) relationships being verified live against −b/a, c/a, −d/a.