What makes an expression a polynomial
Whole powers only
In a polynomial the variable appears only with whole-number powers: $x$, $x^2$, $x^3$, …
↳ A variable under a root or in a denominator is the giveaway.
📚 Chapter Learning Kit
Polynomials · Class 10 Mathematics
English 3 topics 170 leveled MCQs 62 flashcards 3 games Free
Shared by a Veda teacher · Generated with Veda AI · Sep 2026
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Degree
Look for the highest power
The degree is the highest power of the variable, wherever it sits in the expression.
↳ Scan every term before answering.
Linear, quadratic, cubic
Three degrees, three names
↳ The condition a ≠ 0 is part of each definition.
A zero is a place on the graph
Where the curve meets the axis
On the x-axis every point has $y = 0$.
↳ Zeroes are x-coordinates of the crossings.
The graph of a linear polynomial
One line, one crossing
↳ A line meets the axis exactly once.
Which way a parabola opens
Smile or frown
The graph of $ax^2 + bx + c$ is a parabola.
↳ Only the sign of a decides the direction.
📖 Smart notes
What you'll study, topic by topic
1
Polynomials, Degree and Zeroes
A polynomial is an expression built only from whole-number powers of x. Its degree is the highest power; putting a number into it gives a value p(k); and the numbers that make the value 0 are its zeroes. For a linear pol…
- A POLYNOMIAL in x uses only whole-number powers of x: $4x + 2$, $2y^2 - 3y + 4$, $5x^3 - 4x^2 + x - 2$.
- $\frac{1}{x+1}$, $\sqrt{x}$ and $x^{-2}$ are NOT polynomials — the power of x must be a whole number.
- DEGREE = the highest power of the variable. $7u^6 - \frac{3}{2}u^4 + 4u^2 + u - 8$ has degree 6.
~35 min · full explanation, examples & memory tricks in the app
2
What Zeroes Look Like on a Graph
Draw y = p(x) and the zeroes become visible: they are exactly the x-coordinates of the points where the curve meets the x-axis. A line crosses once; a parabola crosses twice, touches once, or misses the axis altogether;…
- The zeroes of p(x) are the x-COORDINATES of the points where the graph of $y = p(x)$ meets the x-axis.
- The graph of a linear polynomial $ax + b$ is a straight line, crossing the x-axis exactly once, at $\left(-\frac{b}{a},\, 0\right)$.
- The graph of a quadratic $ax^2 + bx + c$ is a PARABOLA: it opens UPWARDS when $a > 0$ and DOWNWARDS when $a < 0$.
~35 min · full explanation, examples & memory tricks in the app
3
Zeroes and Coefficients of a Quadratic
You do not have to find the zeroes to know their sum and product. For ax² + bx + c the sum is −b/a and the product is c/a — read straight off the coefficients. Run the idea backwards and you can build a quadratic from an…
- If α and β are the zeroes of $ax^2 + bx + c$ ($a \neq 0$), then $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.
- In words: sum $= \frac{-(\text{coefficient of } x)}{\text{coefficient of } x^2}$, product…
- Mind the sign: the sum carries a MINUS, the product does not.
~45 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
170 questions laddered from warm-up to topper-level, each with an explanation. A taste:
Which of these is a polynomial?
Beginner
A $x^2 - 3x + 2$
B $\frac{1}{x} + 5$
C $\sqrt{x} + 1$
D $x^{-3} + x$
Show answer & explanation
$x^2 - 3x + 2$
Only the first uses whole-number powers of x throughout.
The degree of $5x^3 - 4x^2 + x - 2$ is
Beginner
A 2
B 3
C 5
D 4
Show answer & explanation
3
The highest power of x is 3.
A polynomial of degree 2 is called
Beginner
A linear
B cubic
C constant
D quadratic
Show answer & explanation
quadratic
Degree 1 is linear, 2 quadratic, 3 cubic.
The general form of a linear polynomial is
Beginner
A $ax^2 + bx + c$
B $ax^3 + b$
C $\frac{a}{x} + b$
D $ax + b,\ a \neq 0$
Show answer & explanation
$ax + b,\ a \neq 0$
Degree 1 with a non-zero coefficient of x.
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