Class 10 Polynomial Formulas

Polynomial equations mean the relation between numbers and variables are explained in a pattern.

What is a Polynomial?

Polynomial is an algebraic expression, in which the variables have non-negative powers. A polynomial function is an equation, which consists of a single independent variable, where the variable can occur in the equation more than one time with different degrees of the exponent. A polynomial expression is the one, which has more than two algebraic terms. As the name suggests, Polynomial is a repetitive addition of a monomial or a binomial.

Types of Polynomial

Polynomial equation is of four types :

• Monomial: It may be define as the polynomial which contains only one term. For example, x² , x, y,v, 2y, 5z etc.
• Binomial: The polynomial which contains only two terms is called Binomial. For example, x² – 12x
• Trinomial: The polynomial which contains three terms. For example, x² – 11x+12
• Quadratic Polynomial: It is type of polynomial which contains four terms. For example, x³+23x² – 15x-12

The general Polynomial Formula is,

F(x) = a_nxⁿ + bx^n-1 + a_n-2x^n-2 + …….. + rx +s

• If n is a natural number: aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b +…+ b^n-2a + b^n-1)
• If n is even (n = 2a): xⁿ + yⁿ = (x + y)(x^n-1 – x^n-2y +…+ y^n-2x – y^n-1)
• If n is odd number: xⁿ + yⁿ = (x + y)(x^n-1 – x^n-2y +…- y^n-2x + y^n-1)

Type of Polynomials

• Linear Polynomials: ax + b = 0
• Quadratic Polynomials: ax^2 + bx + c = 0
• Cubic Polynomials: ax^3 + bx^2 + cx + d = 0

Polynomial Identities

• (x + y)²= x² + 2xy + y²
• (x – y)²= x² – 2xy + y²
• x²– y² = (x + y)(x – y)
• (x + a)(x + b) = x²+ (a + b)x + ab
• (x + y + z)²= x² + y² + c² + 2xy + 2yz + 2zx
• (x + y)³= x³ + y³ + 3xy (x + y)
• (x – y)³= x³ – y³ – 3xy (x – y)
• x³+ y³ = (x + y)(x² – xy + y²)
• x³– y³ = (x – y)(x² + xy + y²)
• x³+ y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
• ax² + bx + c = 0 then x = \frac{( -b \pm \sqrt{(b² – 4ac)} )}{2a}