Page 1 Polynomials Polynomial: An algebraic expression of the form p(x) = a 0 + a 1 x + a 2 x 2 + --- + a n x n where a n ?0 is called a polynomial in variable x of degree n. Where a 0 , a 1 , ----- a n are real numbers and each power of x is a non-negative integer. Example:- 2x 2 – 5x + 1 is a polynomial of degree 2. Note: vx + 3 is not a polynomial. Degree of a Polynomial: The highest power of x in a polynomial p(x) is called the degree of polynomial. Different types of Polynomial: Constant Polynomial: A polynomial of degree zero is called a constant polynomial and it is of the form of p(x) = k Linear Polynomial: A polynomial of degree 1 is called a linear polynomial and it is of the form p(x) = ax + b, where a, b are real numbers and a n ?0. For example: 3x – 3 , 5x, etc. Quadratic Polynomial: A polynomial of degree 2 is called a Quadratic Polynomial and it is of the form of p(x) = ax 2 + bx + c , where a, b, c are real numbers and a n ?0. For example: 2x 2 +x - 1 Cubic Polynomial: A polynomial of degree 3 is called a Cubic Polynomial and it is of the form of p(x) = ax 3 + bx 2 + cx + d , where a, b, c, d are real numbers and a n ?0. For example: x 3 – 1 Bi-quadratic Polynomial: A polynomial of degree 4 is called a Bi- quadratic Polynomial and it is of the form of p(x) = ax 4 + bx 3 + cx 2 + dx + e, where a, b, c, d and e are real numbers and a n ?0. Value of a Polynomial: The value of a polynomial p(x) at x = in the given polynomial and is denoted by p(x). Page 2 Polynomials Polynomial: An algebraic expression of the form p(x) = a 0 + a 1 x + a 2 x 2 + --- + a n x n where a n ?0 is called a polynomial in variable x of degree n. Where a 0 , a 1 , ----- a n are real numbers and each power of x is a non-negative integer. Example:- 2x 2 – 5x + 1 is a polynomial of degree 2. Note: vx + 3 is not a polynomial. Degree of a Polynomial: The highest power of x in a polynomial p(x) is called the degree of polynomial. Different types of Polynomial: Constant Polynomial: A polynomial of degree zero is called a constant polynomial and it is of the form of p(x) = k Linear Polynomial: A polynomial of degree 1 is called a linear polynomial and it is of the form p(x) = ax + b, where a, b are real numbers and a n ?0. For example: 3x – 3 , 5x, etc. Quadratic Polynomial: A polynomial of degree 2 is called a Quadratic Polynomial and it is of the form of p(x) = ax 2 + bx + c , where a, b, c are real numbers and a n ?0. For example: 2x 2 +x - 1 Cubic Polynomial: A polynomial of degree 3 is called a Cubic Polynomial and it is of the form of p(x) = ax 3 + bx 2 + cx + d , where a, b, c, d are real numbers and a n ?0. For example: x 3 – 1 Bi-quadratic Polynomial: A polynomial of degree 4 is called a Bi- quadratic Polynomial and it is of the form of p(x) = ax 4 + bx 3 + cx 2 + dx + e, where a, b, c, d and e are real numbers and a n ?0. Value of a Polynomial: The value of a polynomial p(x) at x = in the given polynomial and is denoted by p(x). Graph of Polynomial: Graph of a linear polynomial p(x) = ax + b is a straight line Graph of a quadratic polynomial p(x) = ax 2 + bx + c is a parabola which open upwards like ? if a > 0 Graph of a quadratic polynomial p(x) = ax 2 + bx + c is a parabola which open downwards like n if a < 0 In general, a polynomial p(x) of degree n crosses the x- axis, at most n points. Zeroes of a polynomial: ? is said to be zero of a polynomial p(x) if p(?)= 0. The zeros of polynomial p(x) are actually the X- coordinates of the points where the graph of y = p(x) intersects the X-axis. A polynomial of degree “n” can have at most n zeros. For example: A linear polynomial has only one zero, a quadratic polynomial has two zeroes and a Cubic polynomial has three zeroes. Division Algorithm for Polynomials: 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) x q(x) + r (x) i.e. Dividend = Divisor x 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.Read More

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