Algebraic Expressions and Polynomials
Algebraic expressions are a combination of constants and variables, connected by any or all of the four fundamental operations (+, -, ×, ÷).
Term: Each part of the expression along with its sign.
Monomial: An algebraic expression containing one term. Example: 6a2, 3x2y2
Binomial: An algebraic expression containing two terms. Example: a2 + b2, 7xy + y2
Trinomial: An algebraic expression containing three terms. Example: x2 + y2 + z2, x2 + 2xy + y2
Polynomial: An algebraic expression in which variables do not occur in the denominator, exponents of variables are whole numbers and numerical coefficients of various terms are real numbers. Example: x3 - 2y2+ y - 7
Degree of a polynomial: Sum of the exponents of the variables in a term is called degree of the term. Degree of a polynomial is the same as the degree of its term or terms having the highest degree and non-zero coefficient.
Quadratic polynomial: A polynomial of degree 2. Example: x2 – 3x + 2
Zeros of a polynomial: Values of the variable for which the value of a polynomial is zero.
Special Products: Products like 108 × 108, 97 × 97, 104 × 96 can easily be calculated with the help of (a + b)2, (a – b)2, (a + b)(a –b) respectively. Such products are called special products.
Factorization of polynomials: Factorization of polynomials is a process of writing the polynomial as a product of two (or more) polynomials. Each polynomial in the product is called a factor of the given polynomial.
HCF of polynomials: HCF of two or more polynomials is the product of the polynomials of highest degree and greatest numerical coefficient each of which is a factor of each of the given polynomials.
LCM of polynomials: LCM of two or more polynomials is the product of the polynomials of the lowest degree and the smallest numerical coefficient which are multiples of the corresponding elements of each of the given polynomials.