## Polynomials

*CBSE Class 9 - Maths Chapter 2*

**Algebraic Expression**. For e.g. x² – 7x + 2, xy² – 3 etc.

② A Polynomials p(x) in one variable x is an algebric expression in x of the form

p(x) = a

_{n}x

^{n}+ a

_{n–1}x

^{n–1}+ ... + a

_{1}x + a

_{0}, where a

_{0}, a

_{1}, a

_{2}..... a

_{n}are

**constant**and a n ≠ 0 are called

**coefficients**and

**n**is a positive integer.

③ The highest power of variable x in a polynomial p(x) is called the

**degree**of the polynomial.

④ A polynomial of degree 1 is called a

**linear polynomial**. For e.g. 4x + 9 is a linear polynomial in x.

⑤ A polynomial of degree 2 is called a

**Quadratic Polynomial**. For e.g. 7y² – 4y + 15 is a Quadratic polynomial in y.

⑥ A polynomial of degree 3 is called a

**Cubic Polynomial**. For e.g. 3z³ – 7z² + z – 3 is a cubic polynomial in z.

⑦ A polynomial having one term is called

**monomial**having two terms called

**binomial**and having three terms called

**trinomial**.

⑧ A polynomial of

__degree zero__is called

**constant polynomial.**

⑨ For a polynomial p(x) if p(a) = 0 where a is a real number we say that ‘a’ is a

**zero of the polynomial**.

⑩ If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by a linear polynomial x – a, then the remainder is p(a). This is called

**remainder theorem**.

⑪ If p(x) is a polynomial of degree ≥ 1 and ‘a’ is any real number then

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

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

This is called

**factor theorem**.

⑫ A polynomial of degree ‘n’ can have at

**most n zeros**.

⑬ Algebraic Identities:

ⅰ

*(x + y)² = x² + 2xy + y²*

ⅱ

*(x – y)² = x² – 2xy + y²*

ⅲ

*x² – y² = (x – y) (x + y)*

ⅳ

*(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz*

ⅴ

*(x + y)³ = x³ + y³ + 3xy (x + y)*

ⅵ

*(x – y)³ = x³ – y³ – 3xy (x – y)*

ⅶ

*x³ – y³ = (x – y) (x + xy + y )*

ⅷ

*x³ + y³ = (x + y) (x*

*²*

*– xy + y*

*²*

*)*

ⅸ

*x³ + y³ + z³ – 3xyz = (x + y + z) (x + y² + z² – xy – yz – xz)*

⑭ If x + y + z = 0 then,

*x³ + y³ + z³ = 3xyz*.

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