Every polynomial has zeroes — the special inputs that make it equal zero — and those zeroes are quietly encoded in its coefficients. This companion connects the algebra to the graphs, and gives you tools, flashcards and a CBSE-style quiz to test yourself.
xx²x³αβγ
🧮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.
🍔
Think of it like a burger with a strict ingredient list. A polynomial can only ever contain whole-number "scoops" of x — x, x², x³, and so on — mixed with plain numbers. The moment you see x under a square root, or x sitting in a denominator, that's a forbidden ingredient sneaking in, and the expression stops being a polynomial.
4x + 2 → degree 1
2y² − 3y + 4 → degree 2
5x³ − 4x² + x − √2 → degree 3
7u⁶ − (3/2)u⁴ + 4u² + u − 8 → degree 6
🚫These are NOT polynomials. 1/(x−1) — has x in the denominator (a negative power in disguise: (x−1)⁻¹). √x + 2 — has x under a square root (a fractional power, x^(1/2)). 1/(x²+2x+3) — same denominator problem. The rule is strict: only non-negative integer powers of x are allowed.
📛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)
🎰
Like a vending machine. A polynomial p(x) is a little machine: feed it a number x, and it hands back a value. p(2) just means "feed the machine the number 2 and see what comes out." For p(x) = x² − 3x − 4, feeding in 2 gives p(2) = 2² − 3(2) − 4 = −6.
p(x) = x² − 3x − 4
p(2) = 2² − 3×2 − 4 = −6
p(0) = 0² − 3×0 − 4 = −4
Zero of a polynomial. A real number k is called a zero of a polynomial p(x) if p(k) = 0 — in other words, feeding k into the machine produces nothing but zero.
p(x) = x² − 3x − 4
p(−1) = (−1)² − 3(−1) − 4 = 1 + 3 − 4 = 0
p(4) = 4² − 3(4) − 4 = 16 − 12 − 4 = 0
→ −1 and 4 are zeroes of x² − 3x − 4
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:
x
−2
2
y = 2x + 3
−1
7
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:
x
−2
−1
0
1
2
3
4
5
y = x² − 3x − 4
6
0
−4
−6
−6
−4
0
6
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.
⚽
Think of a parabola like the arc of a thrown ball. If you toss a ball and track its height over time, the path traces a parabola. The moments the ball is exactly at ground level (height = 0) are its "zeroes" — usually the moment you release it and the moment it lands.
🔀Three possible shapes for a parabola. Depending on the polynomial, a parabola meets the x-axis in one of three ways: (i) at two distinct points — two real zeroes; (ii) tangentially at one point (the two zeroes have merged into a repeated zero); or (iii) not at all — no real zero. Either way, a quadratic never has more than 2 zeroes.
Two distinct zeroes
One repeated zero (touches once)
No real zero (never touches)
🌊Cubic polynomials: up to 3 crossings
Cubic curves can wiggle more. Take y = x³ − 4x:
x
−2
−1
0
1
2
y = x³ − 4x
0
3
0
−3
0
Zeroes at x = -2, 0, 2
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.
General rule (Remark). For a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at at most n points. So a polynomial of degree n has at most n zeroes — it can have fewer, but never more.
🛣️
Like a winding road crossing a straight highway. Picture the x-axis as a perfectly straight highway. A degree-n polynomial's graph is a road that's allowed to curve and wiggle at most n−1 times — which caps how many times it can possibly cross the straight highway at n. A gentle line (degree 1) can only cross once; a cubic can weave enough to cross up to three times, but doesn't have to.
🔗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.
🔍
Like knowing two people's combined age and product of ages. Classic puzzle: "Two numbers add up to 7 and multiply to 12 — what are they?" You don't need to guess-and-check; sum and product alone pin the numbers down (3 and 4). A quadratic's coefficients quietly encode exactly this sum and product for its zeroes.
Quadratic case: worked example
Take p(x) = 2x² − 8x + 6. Splitting the middle term:
Sum of zeroes = 1 + 3 = 4 = −(−8)/2 = −(coefficient of x) / (coefficient of x²)
Product of zeroes = 1 × 3 = 3 = 6/2 = (constant term) / (coefficient of x²)
This isn't a coincidence — it's always true. For any quadratic ax² + bx + c with zeroes α and β:
α + β = −b/a
αβ = c/a
Example: x² + 7x + 10
x² + 7x + 10 = (x + 2)(x + 5) → zeroes are −2 and −5
sum = −2 + (−5) = −7 = −(7)/1 ✓
product = (−2)×(−5) = 10 = 10/1 ✓
Example: x² − 3 (irrational zeroes)
Using the identity a² − b² = (a−b)(a+b):
x² − 3 = (x − √3)(x + √3) → zeroes are √3 and −√3
sum = √3 + (−√3) = 0 = −(0)/1 ✓ (there's no x term, so b = 0)
product = √3 × (−√3) = −3 = −3/1 ✓
💡The sum/product relationship holds no matter what kind of zeroes the polynomial has — whole numbers, fractions, or irrationals like √3. It's a structural fact about how factoring works, not a coincidence of "nice" numbers.
🏗️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:
α + β = −3 = −b/a, αβ = 2 = c/a
Choosing a = 1 gives b = 3, c = 2
→ x² + 3x + 2 is a quadratic with these zeroes
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:
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
Example: 2x³ − 5x² − 14x + 8
Checking, this cubic's zeroes turn out to be 4, −2, and 1/2. Verifying all three relationships:
sum: 4 + (−2) + 1/2 = 5/2 = −(−5)/2 ✓
sum of pairs: {4×(−2)} + {(−2)×(1/2)} + {(1/2)×4} = −8 − 1 + 2 = −7 = −14/2 ✓
product: 4 × (−2) × 1/2 = −4 = −8/2 ✓
⚠️The most commonly forgotten relationship. Students usually remember the sum (−b/a) and product (−d/a) of a cubic's zeroes, but forget the middle one — the sum of products taken two at a time equals c/a. All three are needed to fully "verify the relationship" in exam answers.
📋2.4 Summary
📛 Polynomials of degree 1, 2 and 3 are called linear, quadratic, and cubic respectively.
🧮 A quadratic polynomial has the form ax² + bx + c, with a, b, c real and a ≠ 0.
📈 The zeroes of p(x) are exactly the x-coordinates where the graph of y = p(x) meets the x-axis.
🎯 A quadratic has at most 2 zeroes; a cubic has at most 3 zeroes.
🔗 For a quadratic with zeroes α, β: α + β = −b/a, and αβ = c/a.
🎲 For a cubic with zeroes α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, and αβγ = −d/a.