Important Formulas: Polynomials - Mathematics (Maths) Class 9

Polynomial

A polynomial is an algebraic expression in which the variables have non-integral exponents only.

Example- 3x²­­+ 4y + 2, 5x³+ 3x² + 4x +2

Degree of a Polynomial

For a polynomial in one variable - The highest exponent on the variable in a polynomial is the degree of the polynomial.
Example: The degree of the polynomial x²+3x+4 is 3, as the highest power of x in the given expression is x².

Types of Polynomials

• Number of terms
• Degree of the polynomial.

Types of Polynomials are based on the Number of terms:

• Monomial Polynomial – A polynomial with one term. Example: 3x, 8x2, 11xy
• Binomial – A polynomial that has two terms. Example: 6x2+x, 3x+4
• Trinomial – A polynomial that has three terms. Example: x2+4x+4

Types of Polynomials are based on the Degree of the Polynomial:

1. Linear Polynomial - It is a polynomial of degree 1. It is of the form ax+ b where a & b are real numbers and a is not equal to 0.
2. Quadratic Polynomial - It is a polynomial of degree 2. It is one of the form ax²+ bx + c where a, b & c are real numbers and a is not equal to 0.
3. Cubic Polynomial - It is a polynomial of degree 3. It is one of the form ax³+ bx² + cx+ d where a, b & c are real numbers and a is not equal to 0.

Zeroes or roots of a polynomial

It is that value of a variable at which polynomial P(x) becomes zero.

Example:  if polynomial P(x) = x³-6x²+11x-6, Putting x = 1 one get P(1) = 0 then 1 is a zero of polynomial P(x).

• 0 may be a zero of a polynomial
• Linear polynomial can have at most one zero
• Quadratic polynomial can have at most two zeroes
• Cubic polynomial can have at most three zeroes

Geometrical Meaning of the Zeroes of a Polynomial

• The zeroes of a polynomial p(x) are the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.
• Number of zeroes of a polynomial is the number of times the graph intersects the x-axis.

Relationship between Zeroes and Coefficients of a Polynomial

1. if α and β are the zeroes of the quadratic polynomial ax² + bx+ c, then
   
2. If are the zeroes of the cubic polynomial ax³ + bx² + cx+ d = 0,
     

Division Algorithm

The division algorithm states that for any given polynomial p(x) and any non-zero polynomial g(x) there are polynomial q(x) are r(x) such that
p(x) = g(x) × q(x) + r(x)
Dividend = Divisor × Quotient + Remainder
Where r(x) = 0 or degree r(x) < degree g(x)

Number of Zeroes

A polynomial of degree n has at most n zeros.

• A linear polynomial has one zero,
• A quadratic polynomial has two zeros.
• A cubic polynomial has three zeros.

Factorisation of Polynomials

Quadratic polynomials can be factorized by splitting the middle term.
For example, consider the polynomial 2x²−5x+3
Splitting the middle term
The middle term in the polynomial is -5x
Sum = -5
Product = 6
Now, -5 can be expressed as (-2) + (-3) and -2 x -3 = 6
Putting the above value in the gven expression
2x²−5x+3 = 2x²−2x−3x+3
Identify the common factor
2x²−2x−3x+3 = 2x(x−1)−3(x−1)
Taking (x−1) as the common factor, this can be expressed as:
2x(x−1)−3(x−1)=(x−1)(2x−3)

Identities of Algebra

(a+b)² = a²+2ab+b²

(a−b)²=a²−2ab+b²

(x+a)(x+b) = x²+(a+b)x+ab

a²−b² = (a+b)(a−b)

a³−b³ = (a−b)(a²+ab+b²)

a³+b³ = (a+b)(a²−ab+b²)

(a+b)³ = a³+3a²b+3ab²+b³

(a−b)³ = a3−3a²b+3ab²−b³