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.

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 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

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 

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)

Simultaneous Linear Equations: Two linear equations solved together.

Elimination method:

1. Make the coefficients equal
2. Add/subtract 
3. Find one variable
4. Substitute to get the other.

Substitution method:

1. Express one variable in terms of the other, 
2. Substitute 
3. Solve 
4. Substitute back.