We are offering TS 10th Class Maths Notes Chapter 3 Polynomials to learn maths more effectively.

## TS 10th Class Maths Notes Chapter 3 Polynomials

→ Polynomials are algebraic expressions constructed using constants and variables.

→ If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x).

→ Zero polynomial: A polynomial of degree zero is called zero polynomial.

E.g.: f(x) = 8,

g(x) = \(\frac{-5}{8}\)

→ Linear polynomial: A polynomial of degree ‘1’ is called linear polynomial.

E.g.: f(x) = 3x + 5, g(y) = 7y – 1, h(z) = 5z – 3

→ Quadratic polynomial: A polynomial of degree ‘2’ is called a quadratic polynomial.

E.g.: f(x) = 5x^{2}, f(x) = 7x^{2} – 5x, f(x) = 6x^{2} – 7x + 5

→ Cubic polynomial: A polynomial of degree ‘3’ is called a cubic polynomial.

E.g.: f(x) = 5x^{3} + 4x^{2} – 3x + 6

→ Polynomial of degree ‘n’ in Standard Form : A polynomial in one variable Y of degree ‘n’ is an expression of the form.

F(x) = 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} , an are real coefficients and a ^ 0.

→ A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.

→ The zero of the linear polynomial ax + b is \(\frac{-b}{a}\).

→ If a and P are the zeroes of the quadratic polynomial ax^{2} + bx + c then

- α + β = \(\frac{-b}{a}=\frac{-(\text { Coefficient of } x)}{\text { Coefficient of } x^2}\)
- αβ = \(\frac{c}{a}=\frac{\text { Constant term }}{\text { Coefficient of } x^2}\)

→ Division Algorithm : Dividend = Divisor × Quotient + Remainder.

→ Value of a polynomial at a given point: If p(x) is a polynomial in x and a is real number then the value obtained by putting x = a in p(a) is called the value of p(x) at x = a.

E.g.: Let p(x) = 5x^{2} – 4x + 2, then its value at x = 2 is given by

p(2) = 5(2)^{2} – 4(2) + 2 = 5(4) – 8 + 2 = 20 – 8 + 2 = 14

Thus, the value of p(x) at x = 2 is 14.

→ Graph of a polynomial: In algebraic or in set theory language, the graph of a polynomial f(a) is the collection (or set) of all points (a, y) where y = f(x)

- Graph of a linear polynomial ax + b is a straight line.
- The graph of quadratic polynomial (ax
^{2}+ bx + c) is U – shaped, called Parabola.- If a > 0 in ax
^{2}+ bx + c, the shape of parabola is opening upwards ‘U’. - If a < 0 in ax
^{2}+ bx + c, the shape of parabola is opening downwards ‘n’.

- If a > 0 in ax

→ How to make a quadratic polynomial with the given zeroes : Let the zeroes of a quadratic polynomial be α & β.

x = α, x = β

Then, obviously the quadratic polynomial is (x – α)(x – β) i.e., x^{2} – (α + β) x + aβ i.e., x^{2} – (sum of the zeroes) x + product of the zeroes.

→ Some useful relations :

- α
^{2}+ β^{2}= (α + β)^{2}– 2αβ - (α – β)
^{2}= (α + β)^{2}– 4αβ - α
^{2}– β^{2}= (α + β)(α – β) = (α + β) \(\sqrt{(\alpha+\beta)^2-4 \alpha \beta}\) - α
^{3}+ β^{3}= (α + β)^{3}– 3αβ (α + β) - α
^{3}– β^{3}= (α – β)^{3}+ 3αβ (α – β)

→ If α, β, γ are the zeroes of the cubic polynomial ax^{3} + bx^{2} + cx + d = 0 then

- α + β + γ = \(\frac{-b}{a}\)
- αβ + βγ + γα= \(\frac{c}{a}\)
- αβγ = \(\frac{-d}{a}\)

Important Formula:

- α + β = \(\frac{-b}{a}=\frac{-\text { coefficient of } x}{\text { coefficient of } x^2}\)
- αβ = \(\frac{c}{a}=\frac{\text { constant term }}{\text { coefficient of } x^2}\)
- Dividend = Divisor × Quotient + Remainder
- α + β + γ = \(\frac{-b}{a}\)
- αβ + βγ + γα = \(\frac{c}{a}\)
- αβγ = \(\frac{-d}{a}\)

Flow Chart

Ex.: (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^{3} + y^{3} + z^{3} – 3xyz = (x + y + z) (x^{2} + y^{2} + z^{2} – xy – yz – zx)

Pavuluri Mailana (11th Century):

- Pavuluri Mailana of 11th century A.D. was a mathematician – poet of repute, who wrote his magnum opus in Telugu (Prosodical form) and named it Sarasangraha Ganitham.
- Sarasangraha Ganitham Mailana, it seems wrote 10 chapters, but only three chapters are available. These contains mostly arithmetic and some elementary algebra dealing with linear and quadratic equations.