More about Polynomials

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Presentation transcript:

More about Polynomials Chapter 6 More about Polynomials

Polynomials A monomial is a number, or a product of a number and one or more variables with non-negative integral exponents. axn is a monomial where n is a non-negative integer, a is a constant, and x is a variable. 3, 2x3, 4x2, … are monomials. A polynomial is a sum of monomials. e.g. x + 3, 3x2 - 4x + 5, x3 - 2x2 + 4x + 1 are polynomials.

Polynomials -5x3 - 2x2 + 4x - 7 Constant term = -7 Degree of polynomial = 3 -5x3 - 2x2 + 4x - 7 Coefficient of x3 = -5 Coefficient of x2 = -2 Coefficient of x = 4

Like Terms and Unlike Terms 2x, -3x, 9x are called like terms. 2x, 3, 9x2 are called unlike terms. Like terms can be added or subtracted to form a new term, but unlike terms cannot. e.g. 2x - 3x + 9x = 8x

Fundamental Operations of Polynomials (7x2 - 2x + 4) - (2x2 + 3x - 2) = 5x2 - 5x + 6 (3x + 4)(x - 2) = 3x(x - 2) + 4(x - 2) = 3x2 - 6x + 4x - 8 = 3x2 - 2x - 8

Division of Polynomials (I) Quotient Divisor Dividend -5x + 4 Remainder 9 3x2 - 2x + 4 = (x + 1)(3x - 5) + 9

Division of Polynomials (II) 1

Division of Polynomials (II) Quotient +1 x + 1 Divisor 3x3 + 0x2 - 2x + 1 Dividend -3x 2 - 2x x + 1 Remainder 3x3 - 2x + 1 = (x + 1)(3x2 - 3x + 1)

Remainder Theorem When a polynomial f(x) is divided by a linear polynomial x - a, the remainder R = f(a). When a polynomial f(x) is divided by a linear polynomial mx - n, the remainder .

Factor Theorem If a linear polynomial mx - n is a factor of polynomial f(x), i.e. f(x) is divisible by mx - n, then . Conversely, if , then the linear polynomial mx - n is a factor of f(x), i.e. f(x) is divisible by mx - n.

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