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Lesson 2 Polinomials and Algebraic Fractions LESSON 2 POLINOMIALS and ALGEBRAIC FRACTIONS

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Lesson 2 Polinomials and Algebraic Fractions LESSON 2 POLINOMIALS and ALGEBRAIC FRACTIONS Lesson 2 Polinomials and algebraic fractions LESSON 2 POLINOMIALS AND ALGEBRAIC FRACTIONS I would advise you Sir, to study algebra, if you are not already an adept in it: your head would be less muddy, and you will leave off tormenting your neighbors about paper and packthread, while we all live together in a world that is bursting with sin and sorrow. Samuel Johnson (1709-1784) British author. 1. POLINOMIAL FRACTIONS Monomial Fractions If there is only one term (monomial) in the numerator and denominator, then common factors may be reduced directly. Example: Reduce to lowest terms: Solution: Look at the "top" and the "bottom" of this fraction. Find a common number that will divide evenly into both 36 and 24 (answer is 12). Also, notice where the larger exponents of x and y are located. It may help to expand the fraction to see what the exponents are doing. After reducing by a common constant factor of 12 and leaving "leftover" variables on the "top" and "bottom", the answer is: 1 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions Remember: There will always be a larger number (coefficient) "leftover" where there was a larger number "to begin with". Similarly, there will be variables "leftover" where there were "more" (larger exponents) to begin with. The "leftover" variables are determined by subtracting their exponents. 1 Practise with monomial fractions. Division of polinomials It is similar to usual numbers division. Example: Calculate the quotient and remainder of (x3+2x2-4x+3) ÷ (x2-1) Start by rewriting the problem in long division notation. Make sure there are no missing exponents in either the divisor or dividend. In this problem the divisor has no x term; you should write it with a coefficient of 0. 2 x3+2 x 2 − 4 x + 3 x+0 x − 1 1. Divide the first term of the numerator by the first term of the denominator. That x3 = x will be the first term of the quotient: x2 . 2. Then, multiply the quotient by all the terms of the denominator, and subtract. x3+2 x 2 − 4 x + 3 x2 +0 x − 1 −x3 +0 x 2 + x x 2 2x− 3 x + 3 1 http://www.regentsprep.org/Regents/math/ALGEBRA/AV5/Preducefrac.htm 2 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions 2x2 3. Repeat the same with the remainder, dividing = 2 x2 x3+2 x 2 − 4 x + 3 x2 +0 x − 1 −x3 +0 x 2 + x x + 2 2 2x− 3 x + 3 −2 + + 2x 0 x 2 − + 3x 5 Ruffini's rule. Synthetic division or Ruffini's rule is a simplified method of dividing a polynomial p(x) by x-a, where a is any assigned number. In order to explain the steps to implement Ruffini's rule, an example division will be used throughout the explanation: (x4 − 3x2 + 2 ) : (x − 3) 1. If the polynomial is not complete, complete it by adding the missing terms with zeros. 2. Set the coefficients of the dividend in one line. 3. In the bottom left, place the opposite of the independent term of the divisor. 4. Draw a line and lower the first coefficient. 5. Multiply this coefficient by the divisor and place it under the following term. 6. Add the two coefficients. 3 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions 7. Repeat the process above. Repeat the process: Repeat, again: 8. The last number obtained, 56, is the remainder. 9. The quotient is a polynomial of lower degree and whose coefficients are the ones obtained in the division. x3 + 3 x2 + 6x +18 Example Divide by Ruffini's rule: (x5 − 32) : (x − 2) C(x) = x4 + 2x3 + 4x2 + 8x + 16 R = 0 Another example: 2 2 http://www.emathematics.net/videos.php?id=67 4 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions 2. APPLICATIONS OF RUFFINI'S RULE Divisibility criterium Studying Ruffini's Rule, you can see that: If a polynomial P(x) is divisible by (x-a), then a is a divisor of the independent term of the polinomial. The remainder theorem: "If a polynomial P(x) is divided by a binomial (x - a), then the remainder is equal to the numerical value of the polynomial when x = "a", and can be expressed as P(a)". Example: Calculate the remainder of the division P(x) : Q(x) P(x)= x4 − 3x2 + 2 Q(x) = x − 3 P(3) = 34 − 3 · 32 + 2 = 81 − 27 + 2 = 56 Here is a proof of this theorem. 3. FACTORIZING POLINOMIALS Roots of a polinomial If P(a) = 0, then the value x = a is called a root or zero of P(x). When we manage to express a polynomial as a product of linear binomials, or with at least one binomial of this type, we can refer to this as "factorizing the polynomial". Calculate the Roots of the Polynomial: P(x) = x2 − 5x + 6 P(2) = 22 − 5 · 2 + 6 = 4 − 10 + 6 = 0 P(3) = 32 − 5 · 3 + 6 = 9 − 15 + 6 = 0 x = 2 and x = 3 are roots or zeros of the polynomial: P(x) = x2 − 5x + 6, because P(2) = 0 and P(3) = 0. Properties of the Roots and Factors of a Polynomial 1.The zeros or roots are divisors of the independent term of the polynomial. 2. For each root type x = a corresponds to it a binomial of the type (x − a). 5 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions 3. A polynomial can be expressed in factors by writing it as a product of all the binomials of type (x − a), which will correspond to the roots, x = a. x2 − 5x + 6 = (x − 2) · (x − 3) 4. The sum of the exponents of the binomial must be equal to the degree of the polynomial. 5. All polynomials that do not have an independent term accept x = 0 as a root. x2 + x = x · (x + 1) Roots: x = 0, and x = − 1 Example: Find the Roots and Factor the Following Polynomial: Q(x) = x2 - x - 6 The divisors of the independent term are: ±1, ±2, ±3. Q(1) = 12 − 1 − 6 ≠ 0 Q(−1) = (−1)2 − (−1) − 6 ≠ 0 Q(2) = 22 − 2 − 6 ≠ 0 Q(−2) = (−2)2 − (−2) − 6 = 4 +2 +6 = 0 Q(3) = 32 − 3 − 6 = 9 − 3 − 6 = 0 The roots are: x= −2 and x = 3. Q(x) = (x + 2 ) · (x − 3) 4. DIVISIBILITY A polinomial D(x) is a factor or divisor of P(x) if the division P(x):D(x) is exact. In this case, P(x) is a multiple of D(x). A polynomial is called irreducible or prime when it cannot be decomposed into factors. You can get the Least Common Multiple and the Greatest Common Factor the same way you do with numbers. Example: x2 − 1 = (x + 1) · (x − 1) x2 + 3x + 2 = (x +1 ) · (x + 2) lcm (x2 − 1, x2 + 3x + 2) = (x + 1) · (x − 1) · (x + 2) gcf (x2 − 1, x2 + 3x + 2) = (x + 1) 6 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions 5. ALGEBRAIC FRACTIONS An algebraic fraction is the quotient of two polynomials and is represented as: P(x) is the numerator and Q(x), the denominator. Simplification of Algebraic Fractions To simplify an algebraic fraction, divide the numerator and the denominator of the fraction by a polynomial that is a common factor of both. Equivalent fractions Two algebraic fractions are equivalent if: – you can get one of them by simplifying the other, or – you get the same fraction by simplifying both of them If two algebraic fractions: are equivalent, they are represented as: In this case: P(x) · S(x) = Q(x) · R(x). are equivalent because: (x+2) · (x+2) = x2 − 4 By multiplying the numerator and denominator of a given algebraic fraction by the same nonzero polynomial, the resulting algebraic fraction is equivalent to that given. Reducing to a common denominator To reduce two algebraic fractions to a common denominator, find two equivalent algebraic fractions with the same denominator. Example: 7 IES UNIVERSIDAD LABORAL Mercedes López Lesson 2 Polinomials and algebraic fractions 1. Decompose the denominators to find the least common multiple, which will be the common denominator. x2 − 1 = (x + 1) · (x − 1) x2 + 3x + 2 = (x +1 ) · (x + 2) lcm (x2 − 1, x2 + 3x + 2) = (x + 1) · (x − 1) · (x + 2) 2.You can get the lcm by multipling each denominator by one or several different factors. Multiply each numerator by these corresponding factors. Operations You can add, subtract, multiply and divide algebraic fractions the same way you do with numerical fractions. Solve the following exercises in your notebook. Then, check the solutions. 8 IES UNIVERSIDAD LABORAL Mercedes López.
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