Polynomials Class 9 Notes Maths - Chapter 2
Chapter: 2
What Are Polynomial ?
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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
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)
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