Page 1 Polynomials Page 2 Polynomials POLYNOMIALS 5 a n x n + a n - 1 + a x n - 1 + .......+ a 2 x 2 1 x + a 0 “Polynomial is an algebraic expression with power as whole numbers.” Polynomial 5 x + 5 2x 2 2x 3 + 5x 2 - 3 x 4 + 3x 2 - 2x + 1 2x 5 + 2x 3 - 3x 2 - 2x + 6 Degree 0 1 2 3 4 5 Name using number of terms Monomial Binomial Monomial Trinomial Polynomial of 4 terms Polynomial of 5 terms No of terms 1 2 1 3 4 5 Linear polynomial has only one zero Relation between zeros & Coecients of polynomial (1) Linear polynomial (ax + b) Zero : a = -b a -b a c a Real zero exists when graph cuts x-axis (point of intersection) (2) Quadratic polynomial (ax 2 + bx + c) Sum of zeros ( a + ß) = Coecient of middle term (x) Coecent of x 2 Product of zeros ( aß) = Constant term (3 rd term) Coecient of x 2 { { } } Quadratic polynomial has two zeros. y x 1 2 3 4 5 6 y x y x y x y x y x 2 cuts on x - axis 2 real roots 3 cuts on x - axis 3 real roots 1 touch on x - axis 1 root repeated twice ex : (x + 3) 2 1 touch and 1 cut on x - axis. 3 zeros(roots) out of which one repeated twice No cuts on x - axis 2 non real roots as it is a graph of Quadratic Polynomial 4 cuts on x - axis 4 real roots (zeros) Algebraic Identity a 2 + b 2 = (a + b) 2 - 2ab Quadratic polynomial in root (zeros) form x 2 - ( a + ß) x + aß = 0 Polynomials ; where coecients cannot be Zero Page 3 Polynomials POLYNOMIALS 5 a n x n + a n - 1 + a x n - 1 + .......+ a 2 x 2 1 x + a 0 “Polynomial is an algebraic expression with power as whole numbers.” Polynomial 5 x + 5 2x 2 2x 3 + 5x 2 - 3 x 4 + 3x 2 - 2x + 1 2x 5 + 2x 3 - 3x 2 - 2x + 6 Degree 0 1 2 3 4 5 Name using number of terms Monomial Binomial Monomial Trinomial Polynomial of 4 terms Polynomial of 5 terms No of terms 1 2 1 3 4 5 Linear polynomial has only one zero Relation between zeros & Coecients of polynomial (1) Linear polynomial (ax + b) Zero : a = -b a -b a c a Real zero exists when graph cuts x-axis (point of intersection) (2) Quadratic polynomial (ax 2 + bx + c) Sum of zeros ( a + ß) = Coecient of middle term (x) Coecent of x 2 Product of zeros ( aß) = Constant term (3 rd term) Coecient of x 2 { { } } Quadratic polynomial has two zeros. y x 1 2 3 4 5 6 y x y x y x y x y x 2 cuts on x - axis 2 real roots 3 cuts on x - axis 3 real roots 1 touch on x - axis 1 root repeated twice ex : (x + 3) 2 1 touch and 1 cut on x - axis. 3 zeros(roots) out of which one repeated twice No cuts on x - axis 2 non real roots as it is a graph of Quadratic Polynomial 4 cuts on x - axis 4 real roots (zeros) Algebraic Identity a 2 + b 2 = (a + b) 2 - 2ab Quadratic polynomial in root (zeros) form x 2 - ( a + ß) x + aß = 0 Polynomials ; where coecients cannot be Zero Division Algorithm = × + Divide x 4 + 6x 3 - 2x 2 + 7x + 4 by x 2 + 2x + 1 x 2 + 2x +1 ) x 4 + 6x 3 - 2x 2 + 7x + 4 x 4 + 2x 3 + x 2 + 0 + 0 4x 3 - 3x 2 + 7x + 4 x 2 x 2 + 2x + 1 ) 4x 3 - 3x 2 + 7x + 4 4x 3 + 8x 2 + 4x + 0 -11x 2 + 3x + 4 4x x 2 + 2x + 1 ) -11x 2 + 3x + 4 -11x 2 - 22x - 11 25x + 15 -11 Arrange the polynomial in the decreasing order of their powers (Higher to Lower) Step 1 Step 2 Just look at 1 st term of divisor and think of a term that when multiplied with it gives the 1 st terms of dividend Step 4 Repeat the same procedure until we get terms with order of degree lower than that of divisor. In this case ( x 2 ) Step 3 Subtract and get the remainder (take care of signs) - - - - - - - - + + + Dividend Divisor Quotient Remainder = × + x 2 + 4x - 11 25x + 15 x 4 + 6x 3 - 2x 2 + 7x + 4 x 2 + 2x + 1 Numbers of zeros of a polynomial depend on the degree of that polynomial. A Quadratic polynomial can have most 2 zeros and cubic can have at most 3 zeros. ? ? PLEASE KEEP IN MIND Zero of a polynomial is a number/ value of x by which the value of the polynomial becomes zero (0). Zeros are also known as roots. Don’t get confused in this Every polynomial has zeros, sometimes real sometimes non-real. And if the polynomial does not cut the x-axis and still has zeros then they will be in the form of complex numbers (non-real numbers) equal to the degree of the polynomial. ? Degree of dividend should always be greater than the divisor (Division algorithm) . Introduction to polynomials Types and Zeros of Polynomials Scan the QR Codes to watch our free videos ? POLYNOMIALS 6Read More

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