Students can go through AP Board 9th Class Maths Notes Chapter 2 Polynomials and Factorisation to understand and remember the concepts easily.

## AP State Board Syllabus 9th Class Maths Notes Chapter 2 Polynomials and Factorisation

→ An algebraic expression ¡n which the variables involved have only non-negative integral powers is called a polynomial.

E.g.: 5x^{3} – 2x + 8

→ Polynomials can be classified as monomials. binomials, trinomials and polynomials based on the number of terms it contains.

→ A polynomial with a single term is a monomial.

E.g.: 2x, -5x^{2}, \(\frac{6}{7}\)x^{3} etc.

→ A polynomial with two terms is a binomial.

E.g.: 2x + 5y; -3x^{2} + 5x etc.

→ A polynomial with three terms is a trinomial.

E.g.: 3x^{2} + 5x – 8; 3x + 2y – 5z etc.

→ In general a polynomial may contain two or more than two terms.

E.g.: 5x + 8x^{2} – 3x^{3} + 7

→ Degree of a polynomial ¡s the heighest degree of its variable terms.

E.g.: Degree of 3x^{2} + 4xy^{3} + y^{2} is 4.

→ Degree of a constant term is considered as zero.

E.g.: Degree of 4 is zero.

→ The general form of a polynomial is a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} …….. + a_{n-1}x + a_{n} where a_{0}, a_{1}, a_{2},…… a_{n-1}, a_{n} are constants and a_{0} ≠ 0. Its degree is ‘n’.

→ Polynomials are again classified based on their degrees.

→ The zero of a polynomial p(x) is the value of x at which p(x) becomes zero (i.e.) p(x) = 0.

E.g.: Zero of 3x – 5 is x = \(\frac{5}{3}\)

→ To find the zero of a polynomial we equate the polynomial to zero and solve for the value of the variable.

E.g.: To find zero of 7x + 8.

7x + 8 = 0

7x = – 8

x = \(\frac{-8}{7}\)

→ Let p(x) be any polynomial of degree greater than or equal to one and let ‘a’ be any real number. If p(x) is divided by the linear polynomial (x – a), then the remainder is p(a). This is called the Remainder theorem.

E.g.: If p(x) = 4x^{3} + 3x + 8 then the remainder when it is divided by x – 1 is p(1).

i.e., p(1) = 4 + 3 + 8 = 15.

→ If p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then

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

(ii) and its converse “if (x – a) is a factor of a polynomial p(x) then p(a) = 0. This is called Factor theorem”.

E.g.: Let p(x) = x^{2} – 5x + 6 and p(2) = 2^{2} – 5(2)+ 6 = 0 and hence (x – 2) is a factor of p(x) conversely; p(x) = x^{2} + 7x + 10 and (x + 2) is a factor, then p(-2) = 0.

→ Algebraic identities

- (x + y)
^{2}= x^{2}+ 2xy + y^{2} - (x – y)
^{2}= x^{2}– 2xy + y^{2} - (x + y) (x-y) = x
^{2}– y^{2} - (x + a) (x + b) = x
^{2}+ (a + b) x + ab - (x + y + z)
^{2}= x^{2}+ y^{2}+ z^{2}+ 2xy + 2yz + 2zx - (x + y)
^{3}= x^{3}+ y^{3}+ 3xy (x + y) - (x – y)
^{3}= x^{3}– y^{3}– 3xy (x – y) - (x + y + z) (x
^{2}+ y^{2}+ z^{2}– xy – yz – zx) = x^{3}+ y^{3}+ z^{3}– 3xyz