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 – 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 `k’ is called a zero of a polynomial p(x), if p(K) = 0

• If `k’ is the zero of p(x), then we can also say `k’ 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 ≥ 1) and p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

Factor Theorem

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

Algebraic Identities

• (x + y)2 = x2 + y2 + 2xy
• (x – y)2 = x2 + y2 – 2xy
• x2 – y2= (x + y) (x – 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 – y)3 = x3 – y3 – 3xy (x – y) = x3 – y3 – 3x2y + 3xy2
• x3 + y3 = (x + y) (x2 + y2 – xy)
• x3 – y3 = (x – y) (x2 + y2 + xy)
• x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)