Introduction and Types of Polynomials

What is a Polynomial?

Let's start simple. You have already worked with expressions like $2x + 3$, $x^2 - 5x + 6$, and $4x^3 + x$. These all belong to one family called polynomials.

A polynomial in one variable $x$ is an expression of the form

$$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$

where $a_0, a_1, \dots, a_n$ are real numbers (the coefficients), $a_n \neq 0$, and the powers of $x$ are whole numbers (0, 1, 2, …).

Key Point: The powers of the variable in a polynomial must be non-negative integers. So $x^2 + 3x + 1$ is a polynomial, but $x^{-1} + 2$ or $\sqrt{x} + 1$ (i.e. $x^{1/2}+1$) are not polynomials.

Think of it this way: a polynomial is a 'clean' algebraic expression — no variables in the denominator, no roots of the variable, no negative powers.

[Board Important] A favourite trap: $\dfrac{1}{x} + x$ is not a polynomial because $\dfrac{1}{x} = x^{-1}$ has a negative power.

Terms, Coefficients and Degree

Let's break down the parts of a polynomial using $p(x) = 4x^3 - 2x^2 + 7x - 5$.

Terms

The parts separated by + or − signs: $4x^3$, $-2x^2$, $7x$, and $-5$ are the four terms.

Coefficients

The number multiplying each power of $x$. Here the coefficient of $x^3$ is 4, of $x^2$ is $-2$, of $x$ is 7, and the constant term is $-5$.

Degree

The degree of a polynomial is the highest power of the variable that appears (with non-zero coefficient). For $p(x) = 4x^3 - 2x^2 + 7x - 5$, the degree is 3.

Key Point: Degree decides almost everything about a polynomial — its name, the shape of its graph, and the maximum number of zeroes it can have.

[Board Important] The constant polynomial like $p(x) = 5$ has degree 0. The zero polynomial $p(x) = 0$ has no defined degree.

Types of Polynomials by Degree

We name polynomials according to their degree. These three are the stars of Class 10.

Degree	Name	General form	Example
1	Linear	$ax + b,\ a\neq 0$	$2x + 3$
2	Quadratic	$ax^2 + bx + c,\ a\neq 0$	$x^2 - 5x + 6$
3	Cubic	$ax^3 + bx^2 + cx + d,\ a\neq 0$	$x^3 - 1$

The word quadratic comes from 'quadra' meaning square (because of the $x^2$ term), and cubic from the cube ($x^3$).

Key Point: A linear polynomial has at most 1 zero, a quadratic at most 2 zeroes, and a cubic at most 3 zeroes. In general, a polynomial of degree $n$ has at most $n$ zeroes.

[Board Important] You can also classify by number of terms: monomial (1 term), binomial (2 terms), trinomial (3 terms). But degree-based naming is far more important for this chapter.

Value and Zero of a Polynomial

This idea is the heart of the whole chapter.

Value of a polynomial

The value of $p(x)$ at $x = k$ is the number you get by substituting $x = k$, written $p(k)$.

For $p(x) = x^2 - 5x + 6$: $p(2) = (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$.

Zero of a polynomial

A real number $k$ is a zero of $p(x)$ if $p(k) = 0$. So $x = 2$ is a zero of $x^2 - 5x + 6$ because $p(2) = 0$.

Key Point: Finding a zero means solving $p(x) = 0$. The zeroes of a polynomial are exactly the solutions (roots) of the equation $p(x) = 0$.

Zero of a linear polynomial

For $p(x) = ax + b$, set $ax + b = 0$ to get the single zero $x = -\dfrac{b}{a}$.

[Board Important] Don't confuse 'zero of a polynomial' (a value of $x$ making $p(x)=0$) with 'the number zero'. They are different ideas with the same name.

Solved Examples

Example 1: Is it a polynomial?

Which of these are polynomials: (i) $3x^2 - 4x + 1$, (ii) $x + \dfrac{1}{x}$, (iii) $\sqrt{x} + 2$?

Solution:

1. (i) Powers of $x$ are 2, 1, 0 — all whole numbers → polynomial.
2. (ii) $\dfrac{1}{x} = x^{-1}$ has a negative power → not a polynomial.
3. (iii) $\sqrt{x} = x^{1/2}$ has a fractional power → not a polynomial.

Final Answer: Only (i) is a polynomial.

Takeaway: Check that every power of $x$ is a non-negative integer.

Example 2: Find the degree

Find the degree of $p(x) = 7 - 3x + 5x^4 - x^2$.

Solution:

1. Identify all powers of $x$: $x^4$, $x^2$, $x^1$, $x^0$.
2. The highest power with a non-zero coefficient is $x^4$.

Final Answer: Degree $= 4$.

Takeaway: Order doesn't matter — scan for the highest power present.

Example 3: Classify by degree

Classify as linear, quadratic or cubic: (i) $5x + 2$, (ii) $3x^2 - 7$, (iii) $x^3 + 2x - 1$.

Solution:

1. (i) Degree 1 → linear.
2. (ii) Degree 2 → quadratic.
3. (iii) Degree 3 → cubic.

Final Answer: Linear, quadratic, cubic respectively.

Takeaway: Degree 1 = linear, 2 = quadratic, 3 = cubic.

Example 4: Find the value

If $p(x) = 2x^2 - 3x + 1$, find $p(1)$ and $p(-2)$.

Solution:

1. $p(1) = 2(1)^2 - 3(1) + 1 = 2 - 3 + 1 = 0$.
2. $p(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15$.

Final Answer: $p(1) = 0$, $p(-2) = 15$.

Takeaway: Substitute carefully — watch the signs when the input is negative.

Example 5: Check whether a number is a zero

Is $x = 3$ a zero of $p(x) = x^2 - 2x - 3$?

Solution:

1. Compute $p(3) = (3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0$.
2. Since $p(3) = 0$, yes, 3 is a zero.

Final Answer: Yes, $x = 3$ is a zero.

Takeaway: A number is a zero if substituting it gives 0.

Example 6: Zero of a linear polynomial

Find the zero of $p(x) = 3x - 12$.

Solution:

1. Set $p(x) = 0$: $3x - 12 = 0$.
2. $3x = 12 \Rightarrow x = 4$.

Final Answer: The zero is $x = 4$.

Takeaway: For $ax + b$, the zero is $x = -b/a$; here $-(-12)/3 = 4$.

Example 7: Find the zeroes of a quadratic by factorisation

Find the zeroes of $p(x) = x^2 - 5x + 6$.

Solution:

1. Factorise: find two numbers multiplying to $6$ and adding to $-5$: they are $-2$ and $-3$.
2. $x^2 - 5x + 6 = (x - 2)(x - 3)$.
3. Set each factor to 0: $x = 2$ or $x = 3$.

Final Answer: The zeroes are 2 and 3.

Takeaway: Factorise, then set each factor equal to zero.

Example 8: Coefficient and constant term

For $p(x) = -2x^3 + 4x^2 - x + 7$, state the coefficient of $x^2$, the coefficient of $x$, and the constant term.

Solution:

1. Coefficient of $x^2$ is $4$.
2. Coefficient of $x$ is $-1$.
3. Constant term is $7$.

Final Answer: $4$, $-1$, and $7$ respectively.

Takeaway: Read coefficients with their signs attached.

Example 9: Number of zeroes from degree

What is the maximum number of zeroes of (i) a linear, (ii) a quadratic, (iii) a cubic polynomial?

Solution:

1. A degree-$n$ polynomial has at most $n$ zeroes.
2. (i) Linear (degree 1) → at most 1 zero.
3. (ii) Quadratic (degree 2) → at most 2 zeroes.
4. (iii) Cubic (degree 3) → at most 3 zeroes.

Final Answer: 1, 2, and 3 respectively.

Takeaway: Max zeroes = degree of the polynomial.

Example 10: Verify a given zero

Verify that $x = -1$ is a zero of $p(x) = x^3 + 2x^2 - x - 2$.

Solution:

1. $p(-1) = (-1)^3 + 2(-1)^2 - (-1) - 2$.
2. $= -1 + 2(1) + 1 - 2 = -1 + 2 + 1 - 2 = 0$.
3. Since $p(-1) = 0$, $x = -1$ is a zero.

Final Answer: Yes, $x = -1$ is a zero.

Takeaway: Careful substitution of negatives: $(-1)^3 = -1$ and $(-1)^2 = 1$.