Revision: Algebra Mathematics SSLC (English Medium) Class 9 Tamil Nadu Board of Secondary Education

A polynomial is an algebraic expression made up of terms in which the variables have non‑negative whole-number exponents.

The degree of a polynomial is simply the highest exponent (power) in the expression.

Example 1: 4x² - 3x⁵ + 8x⁶

• Term 1: 4x² → exponent = 2
• Term 2: -3x⁵ → exponent = 5
• Term 3: 8x⁶ → exponent = 6
• Degree = 6 (highest exponent)

Example 2: 25 - x⁴

• Term 1: 25 → exponent = 0 (since 25 = 25x⁰)
• Term 2: -x⁴ → exponent = 4
• Degree = 4

When an algebraic expression is made of only one variable, it is called a polynomial in one variable.

Examples of Polynomials in One Variable:

The highest power of the variable in a polynomial is called its degree.

A real number k is a zero of p(x) if p(k) = 0.

A real number k is a zero of p(x) if p(k) = 0.

A polynomial g(x) is called a factor of the polynomial f(x) if g(x) divides f(x) exactly, giving 0 as the remainder.

Identity: An identity is an equality, which is true for all values of the variables in equality.

The HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers.

An equation which contains two variables and the degree of each term containing a variable is one is called a linear equation in two variables.  

General Form:

ax + by + c = 0 

In simple words, all three must hold:

• It has two variables (usually x and y).
• Every variable term has power 1 (no x², xy, 1/x, √x).
• It can be written as ax + by + c = 0, with a and b not both zero.

A system of simultaneous linear equations is a pair of linear equations in the same two variables that must be satisfied together.

The standard form is:

A solution is an ordered pair (x,y) that satisfies both equations.

A real number k is a zero of p(x) if p(k) = 0.

For

p(x) = ax + b

Zero: \[x=-\frac{b}{a}\]​

For

p(x) = ax + b

Zero: \[x=-\frac{b}{a}\]​

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b².

(a + b)(a - b) = a² - b²

• (x + a)(x + b) = x² + (a + b)x + ab

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac.

• (a + b)³ = a³ + 3a²b + 3ab² + b³.

• (a - b)³ = a³ - 3a²b + 3ab² - b³.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b)(a - b) = a² - b²

• a³ + b³ = ( a + b )(a² - ab + b²)

• a³ - b³ = ( a - b )( a² + ab + b²)

For

p(x) = ax + b

Zero: \[x=-\frac{b}{a}\]​

Theorem :If p(x) is a polynomial of degree `n >= 1` and a is any real number, then
(i) x – a is a factor of p(x), if p(a) = 0, and 
(ii) p(a) = 0, if x – a is a factor of p(x).

Proof:  By the Remainder Theorem, p(x)=(x – a) q(x) + p(a).

(i) If p(a) = 0, then p(x) = (x – a) q(x), which shows that x – a is a factor of p(x).

(ii) Since x – a is a factor of p(x), p(x) = (x – a) g(x) for same polynomial g(x).
In this case, p(a) = (a – a) g(a) = 0.

Quadratic polynomial

$,\; a\; \ne \; 0$

Cubic polynomial

$,\; a\; \ne \; 0$

Statement:
If a polynomial f(x) is divided by (x − a), then the remainder is f(a).

Result:
$.$

Statement

If a polynomial f(x) is divided by (x − a) and the remainder is zero, then (x − a) is a factor of f(x).

Result

(x − a) is a factor of f(x)  ⟺  f(a) = 0

To check whether (x − a) is a factor → find f(a)

• If f(a) = 0 → factor

• If f(a) ≠ 0 → not a factor

Important Forms

• (x − a) is a factor ⇔ f(a) = 0

• (x + a) is a factor ⇔ f(−a) = 0

• (ax + b) is a factor ⇔ \[f(-\frac{b}{a})\] = 0

Statement:
On dividing a polynomial f(x)by a polynomial g(x), there exist polynomials q(x) and r(x) such that

f(x) = g(x)q(x) + r(x)

where either r(x) = 0 or degree of r(x) < degree of g(x)

Result:
degree of r(x) < degree of g(x)

• General form: ax + by + c = 0, a and b not both zero.
• Every variable term has degree 1.
• The graph is a straight line.
• To reach the general form, shift all terms to one side.

• Simultaneous equations must be solved together.

• A solution must satisfy both equations.

• Substitution replaces one variable using an expression from the other equation.

• Elimination removes one variable by adding or subtracting suitably modified equations.