## Polynomials

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*Study Points*

*Study Points*

*. An expression of the form a*

**1**_{0}+ a

_{1}x + a

_{2}x

^{2}+ ----- + a

_{n}x

^{n}where a

_{n}is called a polynomial in variable x of degree n. where; a

_{0},a

_{1}, ----- a

_{n}are real numbers and are called terms/co-efficients of the polynomial and each power of x is a non negative integer.

**. Polynomials in the variable x are denoted by f(x), g(x), h(x) etc.**

*2*e.g. f(x) = a

_{0}+ a

_{1}x + a

_{2}x

^{2}+ ----- + a

_{n}x

^{n}

**. A polynomial p(x) = a (where a is constant) is of degree 0 and is called a constant polynomial.**

*3***. A polynomial p(x) = ax + b is of degree 1 and is called a linear polynomial. e.g. 4x - 3, 5x...**

*4***5**. A polynomial p(x) = ax

^{2}+ bx + c of degree 2 and is called a quadratic polynomial.

e.g. 3x

^{2}-x + 5, 1- x

^{2}...

**. A polynomial p(x) = ax**

*6*^{3}+ bx

^{2}+ cx + d of degree 3 and is called a cubic polynomial.

e.g. √5x

^{3}- 2x

^{2}+ 5x -1...

**. Expressions like 5x**

*7*^{2}+ 1/x , x

^{-2}- x + 1 are not polynomials

**. Zeroes of a polynomial f(x): A real number α is a zero of the polynomial f(x)**

*8*if and only if f(α) = 0. The graph of y = f(α) intersects the X-axis.

**. A linear polynomial has only one zero (at the most).**

*9***. A Quadratic polynonial at the most has two zeroes.**

*10***. A cubic polynomial at the most has three zeroes.**

*11***. A polynomial may not have real zeroes. f(x) ≠ 0 for any value of x. Hence f(x) has no real zeroes.**

*12***. A polynomial of degree n can have at most n real zeroes.**

*13***. For any quadratic polynomial ax**

*14*^{2}+ bx + c = 0, a ≠ 0, the graph of this equation i.e. y = ax

^{2}+ bx + c has one of the two shapes called Parabolas.

**. The parabola for the given quadratic equation will open upwards like ∪ or downwards like ∩ , depending on whether a > 0 or a < 0.**

*15***. If α and β are the zeroes of a quadratic polynomial ax**

*16*^{2}+ bx + c, then

**. A quadratic polynomial whose zeroes are given by α and β is given by:**

*17*p(x) = x

^{2}- (α + β)x + αβ

= x

^{2}- (sum of zeroes)x + product of zeroes.

**. If α, β and γ are zeroes of the cubic polynomial ax**

*18*^{3}+ bx

^{2}+ cx + d then:

α+β+γ = -b/a

αβ + βγ + γα = c/a

αβγ = -d/a

**. The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x),**

*19*there are polynomials q(x) and r(x) such that

p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

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