CBSE Class 9 Math, Polynomials

•  An expression p(x) = a₀xⁿ + a₁x^n–1 + a₂x^n–2 + ... an is a polynomial Where a₀, a₁, .......... an are real numbers and n is non-negative integer.

•  Degree of a polynomial is the greatest exponent of the variable in the polynomial.

•  Constant polynomial is a polynomial of degree zero. The constant plynomial f(x) = 0 is called zero polynomial.

•  Degree of zero polynomial is not defined.

•  A polynomial of degree one is called a linear polynomil e.g. ax + b, where a ≠ 0.

•  A polynomial of degree two is called a quadratic polynomial e.g. ax² + bx + c where a ≠ 0.

•  A polynomial of degree 2 is called a cubic polynomial e.g. px³ + qx² + rx + s, p ≠ 0.

•  A polynomial of degree 4 is called a biquadratic polynomial e.g. px⁴ + qx³ + rx² + sx + t, p ≠ 0.

•  Value of a polynomial p(x) at x – a is p(a).

•  Zero of a polynomial p(x) is a number ‘a’ such that p(a) = 0.

•  Let p(x) is a polynomial of degree greater than or equal to 1 and a is any real number, if p(x0 is divided by the linear polynomial x – a then the remainder is p(a).

•  If p(x) is a polynomial of degree x �d 1 and a is any real number then.

    (i)      x – a is a factor of p(x) if p(a) = 0.

    (ii)    p(a) = 0 if (x – a) is a factor of p(x).

•  (x + y)² = x² + 2xy + y²

•  (x – y)² = x² – 2xy + y²

•  x² – y² = (x + y)(x – y)

•  (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

•  (x + a)(x + b) = x² + (a + b)x + ab

•  (x + y)³ = x³ + y³ + ³xy(x + y)

•  (x – y)³ = x³ – y³ – 3xy(x – y)

•  x³ + y³ + z³ – 3xyz = [(x + y + z) (x² + y² + z² – xy – yz – zx)]

•  If x + y + z = 0, then x³ + y³ + z³ = 3xyz.

•  x³ – y³ = (x + y) (x² + xy + y²)

•  x³ – y³ = (x – y) (x² + xy + y²)

•  The exponent of the term with the highest power in a polynomial is known as its degree. f(x) = 8x³ – 2x² + 8x – 21 and g(x) = 9x² – 3x + 12 are polynomials of degree 3 and 2 respectively.

•  Value of polynomial: The value of a polynomial f(x) at x = c is obtained by substituting x = c in the given polynomial and is denoted by f(c).

•  Zero or root: A real number c is a zero of the polynomial f(x) = a⁰xⁿ + a₁x^n–1 + ... + aⁿ, if f(c) = 0. ⇒ a₀cⁿ + a₁c^n–1 + a₂c^n–2 + ... + a_n = 0