That the division algorithm for polynomials works and gives unique results follows from a simple induction argument on the degree. investigate two algorithms for univariate polynomial arithmetic over Z. According to questions, remainder is x + a ∴  coefficient of x = 1 ⇒  2k  – 9 = 1 ⇒  k = (10/2) = 5 Also constant term = a ⇒  k2 – 8k + 10 = a  ⇒  (5)2 – 8(5) + 10 = a ⇒  a = 25 – 40 + 10 ⇒  a = – 5 ∴  k = 5, a = –5, Filed Under: Mathematics Tagged With: Division Algorithm For Polynomials, Division Algorithm For Polynomials Examples, Polynomials, ICSE Previous Year Question Papers Class 10, Factorization of polynomials using factor theorem, Division Algorithm For Polynomials Examples, Concise Mathematics Class 10 ICSE Solutions, Concise Chemistry Class 10 ICSE Solutions, Concise Mathematics Class 9 ICSE Solutions, Essay on Sociology Topics | Sociology Topics Essay for Students and Children in English, Essay on Agra | Agra Essay for Students and Children in English, Chandrayaan 2 Essay | Essay on Chandrayaan 2 for Students and Children in English, What are the Types of Relations in Set Theory. Slow division algorithms produce one digit of the final quotient per iteration. For example, consider the equation f(x) = 2x 4 – 9x 3 – 21x 2 + 88x + 48, which has the following possible rational roots:. The Division Algorithm for Polynomials Handout Monday March 5, 2012 Let F be a field (such as R, Q, C, or Fp for some prime p). Step 4:Continue this process till the degree of remainder is less t… E.g. Solution: Remainder is 30. Stan- To find Zeroes of Polynomial . So, 3x4 + 6x3 – 2x2 – 10x – 5 = (3x2 – 5) (x2 + 2x + 1) + 0 Quotient = x2 + 2x + 1 = (x + 1)2 Zeroes of (x + 1)2 are –1, –1. Division of a Polynomial by a Polynomial Example4: Using division show that is a factor of . Example 3:    Apply the division algorithm to find the quotient and remainder on dividing p(x) by g(x) as given below p(x) = x4 – 3x2 + 4x + 5, g (x) = x2 + 1 – x Sol. p(x) = g(x) * q(x) + r(x) Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form x−k. Polynomial division refers to performing the division algorithm on polynomials instead of integers. Division with polynomials (done with either long division or synthetic division) is analogous to long division in arithmetic: we take a dividend divided by a divisor to get a quotient and a remainder (which will be zero if the divisor is a factor of the dividend). 2t4 + 3t3 – 2t2 – 9t – 12 = (2t2 + 3t + 4) (t2 – 3). (i)   Let q(x) = 3x2 + 2x + 6, degree of q(x) = 2 p(x) = 12x2 + 8x + 24, degree of p(x) = 2 Here, deg p(x) = deg q(x) (ii)   p(x) = x5 + 2x4 + 3x3+ 5x2 + 2 q(x) = x2 + x + 1, degree of q(x) = 2 g(x) = x3 + x2 + x + 1 r(x) = 2x2 – 2x + 1, degree of r(x) = 2 Here, deg q(x) = deg r(x) (iii)   Let p(x) = 2x4 + x3 + 6x2 + 4x + 12 q(x) = 2, degree of q(x) = 0 g(x) = x4 + 4x3 + 3x2 + 2x + 6 r(x) = 0 Here, deg q(x) = 0, Example 8:    If the zeroes of polynomial x3 – 3x2 + x + 1 are a – b, a , a + b. 3. Sol. In algebra, an algorithm for dividing a polynomial by another polynomial of the same or lower degree is called polynomial long division. ∴  x = 2 ± √3 ⇒  x – 2 = ±(squaring both sides) ⇒  (x – 2)2 = 3      ⇒   x2 + 4 – 4x – 3 = 0 ⇒  x2 – 4x + 1 = 0 , is a factor of given polynomial ∴  other factors \(=\frac{{{\text{x}}^{4}}-6{{\text{x}}^{3}}-26{{\text{x}}^{2}}+138\text{x}-35}{{{\text{x}}^{2}}-4\text{x}+1}\) ∴  other factors = x2 – 2x – 35 = x2 – 7x + 5x – 35 = x(x – 7) + 5(x – 7) = (x – 7) (x + 5) ∴  other zeroes are (x – 7) = 0 ⇒ x = 7 x + 5 = 0 ⇒  x = – 5, Example 10:     If the polynomial x4 – 6x3 + 16x2 –25x + 10 is divided by another  polynomial x2 –2x + k, the remainder comes out to be x + a, find k & a. Sol. Let us consider the second exercise of the “Polynomial division in practice” step. Transcript. Sol. “. Since two zeroes are \(\sqrt{\frac{5}{3}}\)  and   \(-\sqrt{\frac{5}{3}}\) x = \(\sqrt{\frac{5}{3}}\), x = \(-\sqrt{\frac{5}{3}}\) \(\Rightarrow \left( \text{x}-\sqrt{\frac{5}{3}} \right)\left( \text{x +}\sqrt{\frac{5}{3}} \right)={{\text{x}}^{2}}-\frac{5}{3}\)   Or  3x2 – 5 is a factor of the given polynomial. gcd of polynomials using division algorithm If f (x) and g(x) are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend. Division algorithms fall into two main categories: slow division and fast division. Polynomial Division. So the division algorithm holds. Ask Question Asked 2 days ago. 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