Polynomials Class 9 Notes Maths - Chapter 2

Chapter: 2

What Are Polynomial ?

Meaning Of A Polynomial

  • Polynomial is an algebraic expression, where variables have only whole numbers as an exponent.
    Like x2 โ€“ y2 + 2xy, x2 + 2, etc.
  • A polynomial in one variable, say x, is an algebraic expression of the form P(x) = anxn + an-1 xn-1 +.......a2x2 + a1x + a0 , where a0, a1, a2,.......,an are constants and respectively known as coefficients of x0, x, x2......., xn.

Terms Of A Polynomial

If a polynomial p(x) = anxn + an-1xn-1 + ..... + a2x2 + a1x + a0, then each of anxn, an-1 xn-1,......, a0 is called a term of the polynomial p(x).

Degree Of A Polynomial

degreeofpolynomial

Types Of Polynomials

According to Number of Terms:

  • Polynomials having only one term are called monomials.
  • Polynomials having only two terms are called binomials.
  • Polynomials having only three terms are called trinomials.

According to Degree:

  • A polynomial of degree 0 is called constant polynomial.
  • A polynomial of degree 1 is called linear polynomial.
  • A polynomial of degree 2 is called quadratic polynomial.
  • A polynomial of degree 3 is called cubic polynomial.
  • A polynomial of degree 4 is called biquadratic polynomial.

Zeroes Of A Polynomial

  • Any real number `kโ€™ is called a zero of a polynomial p(x), if p(K) = 0
  • If `kโ€™ is the zero of p(x), then we can also say `kโ€™ is a root of the equation p(x) = 0

Some Important Facts

  • Every linear polynomial in one variable has a unique zero.
  • Every non-zero constant polynomial has no zero.
  • Every real number is a zero of the zero polynomial

Remainder Theorem

If p(x) is any polynomial of degree n (n โ‰ฅ 1) and p(x) is divided by the linear polynomial x โ€“ a, then the remainder is p(a).

Factor Theorem

x โ€“ a is a factor of the polynomial p(x), if p(a) = 0 and vice versa, i.e. if x โ€“ a is a factor of p(x), then p(a) = 0

Algebraic Identities

  • (x + y)2 = x2 + y2 + 2xy
  • (x โ€“ y)2 = x2 + y2 โ€“ 2xy
  • x2 โ€“ y2= (x + y) (x โ€“ y)
  • (x + a) (x + b) = x2 + (a + b) x + ab
  • (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
  • (x + y)3 = x3 + y3 + 3xy (x + y) = x3 + y3 + 3x2y + 3xy2
  • (x โ€“ y)3 = x3 โ€“ y3 โ€“ 3xy (x โ€“ y) = x3 โ€“ y3 โ€“ 3x2y + 3xy2
  • x3 + y3 = (x + y) (x2 + y2 โ€“ xy)
  • x3 โ€“ y3 = (x โ€“ y) (x2 + y2 + xy)
  • x3 + y3 + z3 โ€“ 3xyz = (x + y + z) (x2 + y2 + z2 โ€“ xy โ€“ yz โ€“ zx)