Students can go through AP SSC 10th Class Maths Notes Chapter 3 Polynomials to understand and remember the concepts easily.

## AP State Syllabus SSC 10th Class Maths Notes Chapter 3 Polynomials

→ Polynomial: An algebraic expression in which the variables involved have only non-negative integer power is called a polynomial.

Ex: 2x + 5, 3x^{2} + 5x + 6, -5y, x^{3}, etc.

→ Polynomials are constructed using constants and variables.

→ Coefficients operate on variables, which can be raised to various powers of non negative integer exponents.

, etc. are not polynomials.

→ General form of a polynomial having n^{th} degree is p(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}, a_{2},…… a_{n-1}, a_{n} are real coefficients and a_{0} ≠ 0.

→ Degree of a polynomial:

The exponent of the highest degree term in a polynomial is known as its degree.

In other words, the highest power of x in a polynomial f(x) is called the degree of a polynomial f(x).

Example:

i) f(x) = 5x + \(\frac{1}{3}\) is a polynomial in the variable x of degree 1.

ii) g(y) = 3y^{2} – \(\frac{5}{2}\)y + 7 is a polynomial in the variable y of degree 2.

→ Zero polynomial: A polynomial of degree zero is called zero polynomial that are having only constants.

Ex: f(x) = 8, f(x) = –\(\frac{5}{2}\)

→ Linear polynomial: A polynomial of degree one is called linear polynomial.

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

More generally, any linear polynomial in variable x with real coefficients is of the form f(x) = ax + b, where a and b are real numbers and a ≠ 0.

Note: A linear polynomial may be a monomial or a binomial because linear polynomial f(x) = \(\frac{7}{5}\)x – \(\frac{5}{2}\) is a binomial, whereas the linear polynomial g(x) = \(\frac{2}{5}\) x is a monomial.

→ Quadratic polynomial: A polynomial of degree two is called quadratic polynomial.

Ex: f(x) = 5x^{2}, f(x) = 7x^{2} – 5x, f(x) = 8x^{2} + 6x + 5.

More generally, any quadratic polynomial in variable x with real coefficients is of the form f(x) = ax^{2} + bx + c, where a, b and c are real numbers and a ≠ 0.

Note: A quadratic polynomial may be a monomial or a binomial or a trinomial.

Ex: f(x) = \(\frac{1}{5}\)x^{2} is a monomial, g(x) = 3x^{2} – 5 is a binomial and

h(x) = 3x^{2} – 2x + 5 is a trinomial.

→ Cubic polynomial: A polynomial of degree three is called cubic polynomial.

Ex: f(x) = 8x^{3}, f(x) = 9x^{3} + 5x^{2}

f(y) = 11y^{3} – 9y^{2} + 7y,

f(z) = 13z^{3} – 12z^{2} + 11z + 5.

→ Polynomial of degree ‘n’ in standard form: A polynomial in one variable x of degree n is an expression of the form f(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}, a_{n} are constants and a_{n} ≠ 0.

In particular, if a_{0} = a_{1} = a_{2} = …… = a_{n} = 0 (all the constants are zero; we get the constants zero), we get the zero polynomial which is not defined.

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

Ex : 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(x) is the collection (or set) of all points (x, y) where y = f(x).

i) Graph of a linear polynomial ax + b is a straight line.

ii) The graph of a 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 ‘∪’.

→ If a < 0 in ax^{2} + bx + c, the shape of parabola is opening downwards ‘∩’

→ Relationship between the zeroes and the coefficient of a polynomial:

Note: Formation of a cubic polynomial : Let α, β, and γ be the zeroes of the polynomial.

∴ Required cubic polynomial = (x – α) (x – β) (x – γ).

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

∴ x = α, x = β

Then, obviously the quadratic polynomial is (x – α) (x – β) i.e., x^{2} – (α + β)x 4- ap.

i.e., x^{2} – (sum of the zeroes) x + product of the zeroes.

→ Division Algorithm : If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that, p(x) = g(x) × q(x) + r(x)

i.e., Dividend = Divisor × Quotient + Remainder

where, r(x) = 0 or degree of r(x) < degree of g(x). This result is known as the division algorithm for polynomials.

→ Some useful relations:

α^{2} + β^{2} = (α + β)^{2} – 2αp

(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)