Polynomials - Class 10 Maths

1. Polynomial

An algebraic expression $p(x)$ of the form

General Form of an $n^{\text{th}}$ Degree Polynomial $$p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$$

where $a_n \neq 0$, and $a_0, a_1, \dots, a_n$ are real numbers, is called a polynomial in variable $x$.

Degree: The highest exponent of $x$ in $p(x)$ with a non-zero coefficient is called the degree of the polynomial ($n$).
All exponents of the variable must be non-negative integers (whole numbers: $0, 1, 2, \dots$). Expressions like $x^{-1}$, $\sqrt{x} = x^{1/2}$, or $\frac{1}{x+2}$ are not polynomials.

2. Classification of Polynomials by Degree

Degree 1

Linear Polynomial

A polynomial of degree $1$. Its general form is:

$$p(x) = ax + b \quad (a \neq 0)$$

Has at most $1$ real zero. Its graph is a straight line.

Degree 2

Quadratic Polynomial

A polynomial of degree $2$. Its general form is:

$$p(x) = ax^2 + bx + c \quad (a \neq 0)$$

Has at most $2$ real zeroes. Its graph is a parabola (opening upward if $a > 0$, downward if $a < 0$).

Degree 3

Cubic Polynomial

A polynomial of degree $3$. Its general form is:

$$p(x) = ax^3 + bx^2 + cx + d \quad (a \neq 0)$$

Has at most $3$ real zeroes. Its graph is an S-shaped curve crossing the $x$-axis up to 3 times.

Degree 4

Biquadratic (Quartic)

A polynomial of degree $4$. Its general form is:

$$p(x) = ax^4 + bx^3 + cx^2 + dx + e \quad (a \neq 0)$$

Has at most $4$ real zeroes. Its graph has up to 3 turning points (W-shape or M-shape).

3. Value of a Polynomial

If $p(x)$ is a polynomial in $x$, and if $k$ is any real number, then the real value obtained by replacing $x$ by $k$ in $p(x)$ is called the value of $p(x)$ at $x = k$, denoted by $p(k)$.

Example If $p(x) = 2x^2 - 3x + 5$, then the value of $p(x)$ at $x = 2$ is: $$p(2) = 2(2)^2 - 3(2) + 5 = 8 - 6 + 5 = 7$$

4. Zeroes of a Polynomial

A real number $k$ is said to be a zero of a polynomial $p(x)$ if the value of the polynomial at $x = k$ is zero:

Definition Condition $$p(k) = 0$$

A polynomial of degree $n$ can have at most $n$ real zeroes.
A zero of a polynomial need not be $0$. Conversely, $0$ may be a zero of a polynomial.

5. Geometrical Meaning of the Zeroes of a Polynomial

The real zeroes of any polynomial $y = p(x)$ correspond geometrically to the $x$-coordinates of the points where the curve intersects or touches the $x$-axis (i.e., where $y = 0$).

5.1 Linear Polynomial: $y = ax + b \quad (a \neq 0)$

The graph of a linear polynomial is always a straight line. Since a non-horizontal line intersects the $x$-axis at exactly one point, a linear polynomial has exactly one real zero at $x = -\frac{b}{a}$.

A straight line crosses the $x$-axis at one unique point: $\left(-\frac{b}{a}, 0\right)$.

5.2 Quadratic Polynomial: $y = ax^2 + bx + c \quad (a \neq 0)$

The graph of a quadratic polynomial is a continuous curve called a parabola. It opens upwards if $a > 0$ and downwards if $a < 0$. The discriminant $D = b^2 - 4ac$ dictates whether it cuts, touches, or misses the $x$-axis:

Case 1: $D > 0$ (2 Distinct Zeroes)

Intersects the $x$-axis at two distinct points $A(\alpha, 0)$ and $B(\beta, 0)$.

Case 2: $D = 0$ (1 Coincident Zero)

Vertex touches the $x$-axis at exactly one point ($\alpha = \beta = -b/2a$).

Case 3: $D < 0$ (No Real Zeroes)

Parabola lies entirely above ($a > 0$) or below ($a < 0$) the $x$-axis without touching it.

5.3 Cubic Polynomial: $y = ax^3 + bx^2 + cx + d \quad (a \neq 0)$

The graph of a cubic polynomial is an inflectional S-shaped curve. Since degree 3 is odd, the curve extends to $\pm\infty$ in opposite vertical directions and must cross the $x$-axis at least once. Depending on turning points, it can produce 1, 2, or 3 real zeroes.

3 Distinct Zeroes (e.g., $y = x^3 - 4x$)

Crosses the $x$-axis at three distinct points.

2 Real Zeroes (e.g., $y = x^3 - x^2$)

Touches at one point (repeated root) and cuts at another.

1 Real Zero (e.g., $y = x^3$)

Strictly monotonic or turning points do not cross the axis again.

Fundamental Takeaway: A polynomial of degree $n$ can intersect the $x$-axis at at most $n$ points. Therefore, a polynomial of degree $n$ has at most $n$ real zeroes.

6. Relationship Between Zeroes and Coefficients

(i) Linear Polynomial: $p(x) = ax + b$ ($a \neq 0$)

Setting $p(x) = 0 \implies ax + b = 0 \implies ax = -b \implies x = -\frac{b}{a}$.

Linear Zero Relation $$\text{Zero } (k) = -\frac{b}{a} = -\frac{\text{Constant term}}{\text{Coefficient of } x}$$

(ii) Quadratic Polynomial: $p(x) = ax^2 + bx + c$ ($a \neq 0$)

Let $\alpha$ and $\beta$ be the zeroes of the quadratic polynomial $p(x)$.

Derivation of Quadratic Relations

1. By Factor Theorem, $(x - \alpha)$ and $(x - \beta)$ are factors of $p(x)$. $$ax^2 + bx + c = k(x - \alpha)(x - \beta) \quad (\text{where } k \neq 0 \text{ is a constant})$$

2. Expanding the right-hand side: $$ax^2 + bx + c = k\big[x^2 - (\alpha + \beta)x + \alpha\beta\big] = kx^2 - k(\alpha + \beta)x + k\alpha\beta$$

3. Equating the coefficients of like powers of $x$ on both sides: $$k = a$$ $$-k(\alpha + \beta) = b \implies -a(\alpha + \beta) = b \implies \mathbf{\alpha + \beta = -\frac{b}{a}}$$ $$k\alpha\beta = c \implies a\alpha\beta = c \implies \mathbf{\alpha\beta = \frac{c}{a}}$$

Quadratic Relations & Formulation $$\text{Sum of zeroes: } \alpha + \beta = -\frac{b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$$ $$\text{Product of zeroes: } \alpha\beta = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2}$$ $$\text{Quadratic Polynomial from Zeroes: } p(x) = k\big[x^2 - (\alpha + \beta)x + \alpha\beta\big]$$

(iii) Cubic Polynomial: $p(x) = ax^3 + bx^2 + cx + d$ ($a \neq 0$)

Let $\alpha, \beta, \gamma$ be the zeroes of the cubic polynomial $p(x)$.

Derivation of Cubic Relations

1. By Factor Theorem, $(x - \alpha), (x - \beta),$ and $(x - \gamma)$ are factors of $p(x)$: $$ax^3 + bx^2 + cx + d = k(x - \alpha)(x - \beta)(x - \gamma)$$

2. Expanding the product: $$ax^3 + bx^2 + cx + d = k\Big[x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \beta\gamma + \gamma\alpha)x - \alpha\beta\gamma\Big]$$

3. Comparing coefficients of $x^3, x^2, x,$ and constant terms: $$k = a$$ $$-k(\alpha + \beta + \gamma) = b \implies \mathbf{\alpha + \beta + \gamma = -\frac{b}{a}}$$ $$k(\alpha\beta + \beta\gamma + \gamma\alpha) = c \implies \mathbf{\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}}$$ $$-k\alpha\beta\gamma = d \implies \mathbf{\alpha\beta\gamma = -\frac{d}{a}}$$

Cubic Relations & Formulation $$\text{Sum of zeroes: } \alpha + \beta + \gamma = -\frac{b}{a}$$ $$\text{Sum of products of zeroes taken two at a time: } \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$$ $$\text{Product of zeroes: } \alpha\beta\gamma = -\frac{d}{a}$$ $$\text{Cubic Polynomial: } p(x) = k\big[x^3 - (\alpha+\beta+\gamma)x^2 + (\alpha\beta+\beta\gamma+\gamma\alpha)x - \alpha\beta\gamma\big]$$

(iv) General $n^{\text{th}}$ Degree Polynomial (Vieta's Formulas)

For a general polynomial $p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0$ ($a_n \neq 0$) with roots $\alpha_1, \alpha_2, \dots, \alpha_n$:

Vieta's General Relations $$\sum_{i=1}^n \alpha_i = \alpha_1 + \alpha_2 + \dots + \alpha_n = -\frac{a_{n-1}}{a_n}$$ $$\sum_{1 \le i < j \le n} \alpha_i \alpha_j = \alpha_1\alpha_2 + \alpha_1\alpha_3 + \dots = \frac{a_{n-2}}{a_n}$$ $$\sum_{1 \le i < j < k \le n} \alpha_i \alpha_j \alpha_k = -\frac{a_{n-3}}{a_n}$$ $$\text{Product of all zeroes: } \alpha_1 \alpha_2 \dots \alpha_n = (-1)^n \frac{a_0}{a_n}$$

Sign Pattern Rule: The signs alternate strictly: negative for sum of roots ($1$ at a time), positive for sum of products taken $2$ at a time, negative for products taken $3$ at a time, and $(-1)^n$ for all $n$ roots taken together.

7. Division Algorithm for Polynomials

If $p(x)$ and $g(x)$ are any two polynomials with $g(x) \neq 0$, then there exist unique polynomials $q(x)$ and $r(x)$ such that:

Division Identity $$p(x) = g(x) \times q(x) + r(x)$$ $$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$

Remainder Condition: Where either $r(x) = 0$, or $\text{deg } r(x) < \text{deg } g(x)$.
• If the remainder $r(x) = 0$, then the divisor $g(x)$ is a factor of the dividend $p(x)$.

8. Method: Zero to Divisor (Finding Remaining Zeroes)

A standard higher-degree examination problem asks to find all remaining zeroes of a cubic or biquadratic polynomial when one or two zeroes are already given.

Step-by-Step Procedure

Step 1 (Convert Zeroes to Factors):
• If $\alpha$ is a given zero, then $(x - \alpha)$ is a factor.
• If two zeroes $\alpha$ and $\beta$ (such as conjugate surds $p \pm \sqrt{q}$) are given: $$(x - \alpha)(x - \beta) = x^2 - (\alpha + \beta)x + \alpha\beta \quad \text{is a factor}$$

Step 2 (Form the Divisor $g(x)$): Let $g(x) = (x - \alpha)(x - \beta)$ be the quadratic divisor.

Step 3 (Divide using Polynomial Long Division):
Divide the given higher-degree polynomial $p(x)$ by $g(x)$.
Since $g(x)$ is a factor, the remainder must be identically zero: $r(x) = 0$. $$p(x) = g(x) \cdot q(x)$$

Step 4 (Solve the Quotient):
Set the quotient polynomial to zero: $q(x) = 0$.
Solving $q(x) = 0$ (via factorization or the quadratic formula) reveals all the remaining zeroes of $p(x)$.