UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. Polynomials: A polynomial is an algebraic expression that consists of a sum of several monomials. Remember that a monomial is an algebraic expression as ax n a non negative integer., where a is a real number, and n is The standard form of a polynomial is : P x =a n x n a n 1 x n 1... a 1 x a 0 Here, n denotes the highest power to which x is raised; this highest exponent is called the degree of the polynomial. Thus, in standard form, the highest power term is listed first, and subsequent powers are listed in decreasing order. The monomial a n x n, which is the monomial with the highest exponent of the variable, is called the leading term. The number a 0, which is the term with the exponent zero of the variable, is called the constant term. For instance, the algebraic expression 3x 5 x 3 4x 2 7x 4 is a polynomial: It has five terms: 3x 5, x 3, 4x 2, 7x and 4. The degree is 5, since this is the highest exponent of the variable x. You can say: it is a fifthdegree polynomial. The leading term is 3x 5, while the constant term is 4. Numerical value of a polynomial: Evaluating a polynomial is the same as calculating its numerical value at a given value of the variable: you plug in the given value of x, and figure out what the polynomial is supposed to be. Example: What are the numerical values of the polynomial x=2 and x= 1? P x =x 3 2x 2 3x 5 at the values P 2 P 1 1
Adding and subtracting polynomials: Only like terms (those with identical letters and powers) can be added or subtracted. 3xy and 5xy are like terms. x 3 and 7 x 2 are unlike terms, because the powers of x are not the same. Addition or subtraction of polynomials are achieved by adding or subtracting like terms. Examples: a) Given the polynomials: P x =x 4 5x 3 x 6 and Q x =x 3 4x 2 3x 2, calculate P x Q x and P x Q x. b) Given the polynomials: P x =x 4 4x 3 2x 2 3x 1 and Q x = x 3 5x 2 x 3, calculate P x Q x and P x Q x. 2
Multiplying polynomials: The product of two polynomials is calculated by the multiplication of all monomials of the two polynomials. Example: Given the polynomials: P x Q x. P x =x 2 4x 1 and Q x = x 2 2x 3, calculate Dividing polynomials: To divide two polynomials the degree of the dividend has to be greater or equal than the degree of the divisor. If P(x) is the dividend, Q(x) is the divisor, C(x) is the quotient and R(x) is the remainder: P(x) R(x) Q(x) C(x) P(x)=Q(x) C(x)+R(x) The degree of the remainder is always less than the degree of the divisor. Example: Calculate the quotient and remainder of the divisions: a) x 4 3x 3 4x 2 2x 5 : x 2 2x 3 3
b) x 3 5x 7 : x 2 3 Ruffini's Rule (Synthetic division): Synthetic division (Ruffini's Rule) is a shorthand method of polynomial division in the special case of dividing by a linear factor x a, and it only works in this case. Synthetic division is also use to find zeroes or roots of the polynomial. In mathematics, Ruffini's Rule allows us the rapid division of a polynomial polynomial like x a. The process is shown with the example below: P x by a Example: If we want to work out the division : 3x 3 2x 2 5 : x 1. P x =3x 3 2x 2 5 is the dividend Q x =x 1 is the divisor. The main problem, we first find, is that Q x is not a binomial of the form rewrite it in this way: x a. We must Q x = x 1 Now, we are going to apply the algorithm: 1. Write down the coefficients and a. Note that, as P x doesn't contain a coefficient for x, we write 0: -1 3 2 0-5 2. Pass the first coefficient down: 3 2 0-5 -1 3 4
3. Multiply the last value by a: 3 2 0-5 -1-3 3 4. Add the values: 3 2 0-5 -1-3 3-1 5. Repeat steps 3 and 4 until we finish: 3 2 0-5 -1-3 1-1 3-1 1-6 (remainder) (result coefficients) So, the division 3x 3 2x 2 5 : x 1 has a quotient C x =3x 2 x 1 and a remainder R x = 6. Realize that the quotient is a polynomial of lower degree (one unit less of the degree of the dividend) and the remainder is always a constant term. Examples: Calculate the quotient and remainder of the following divisions, using Ruffini's Rule: a) x 3 5x 2 x 10 : x 2 b) x 4 3x 3 4x 6 : x 2 c) x 4 x 3 x 2 2x 2 : x 1 d) 2x 3 x 1 : x 3 e) x 3 1 : x 1 f) x 4 3x 2 7 : x 2 5
Remainder Theorem: The remainder of the division P x : x a is the numerical value of the polynomial P x when x=a, P a. Examples: 1. Calculate the remainder of the division: x 3 2x 2 3x 4 : x 2 : a) Calculating the division by Ruffini's Rule: b) Using the Remainder Theorem: 2. Calculate the numerical value of the polynomial P x =x 4 x 2 3x 6 when x= 1. a) Using the definition of numerical value: b) Applying the Remainder Theorem: Factor Theorem: If P(a)=0 then x-a is a factor or a divisor of the polynomial P(x). Example: a) Calculate the numerical value of the polynomial P x =x 3 3x 2 5x 6 when x=2. b) Calculate the division: x 3 3x 2 5x 6 : x 2. 6
Roots of a polynomial: A real number a is a root or a zero of a polynomial P(x) if P(a)=0. Properties: If an integer a is a root of a polynomial P(x), this number a will be a factor or divisor of the constant term of P(x). The number of roots of a polynomial is always less or equal than the degree of the polynomial. Factoring Polynomials: Factoring a polynomial is the opposite process of multiplying polynomial. Recall that when we factor a number, we are looking for prime numbers that multiplying together to give the number, for example: 6=2 3, 12=2 2 3. When we factor a polynomial, we are looking for simpler polynomial that can be multiplied together to give us the polynomial we started with. Factoring a polynomial is to write it as a product of polynomials with the lowest possible degree. Factoring polynomials can be done by: Common Factors. Special Products. Ruffini's Rule. Examples: Factorise the following polynomials: a) x 3 7x 6 b) x 3 6x 2 11x 6 7
c) x 3 3x 2 2x d) x 4 1 Your Turn 1. Factorise the following polynomials: a) x 3 2x 2 x 2 b) x 4 6x 3 4x 2 6x 5 8
c) x 4 3x 3 4x d) x 4 2x 3 3x 2 4x 4 e) x 3 2x 2 x 2 f) x 4 x g) x 4 5x 3 6x 2 h) x 4 4x 2 9
Algebraic Fractions: An algebraic fraction (or a rational expression) is a fraction whose numerator and denominator are polynomials. Two algebraic fractions P x Q x and R x S x are equivalent if P x S x =Q x R x. Algebraic fractions behave the same as numerical fractions. So we can simplify, add, subtract, multiply or divide them, using the same rules. Simplifying algebraic fractions: You can simplify algebraic fractions by cancelling common factors in numerator and denominator to reach an equivalent fraction. Examples: Simplify: a) x 2 2x x 2 4x 4 b) 5x 5 5x 10 c) x 3 x 2 9 d) x 2 1 x 2 x 2 e) x 3 x x 2 x f) x 2 4 x 2 4x 4 Adding and subtracting algebraic fractions: a) With the same denominator: You can add or subtract easily, simply add or subtract the numerators and write the sum over the common denominator. Examples: Calculate: a) b) 3x 4 x 3 x 4 x 3 x 2 5 x 1 x2 6 x 1 b) With different denominators: Before you can add or subtract algebraic fractions with different denominators, you must reduce to common denominator (calculate the LCM) and then add or subtract numerators. 10
Examples: Calculate: a) b) x x 2 x 3 x 1 x 1 x 2 4 x x 2 Multiplying and dividing algebraic fractions: P x Q x Examples: Calculate: R x P x R x = S x Q x S x P x Q x : R x P x S x = S x Q x R x a) b) 2x x 3 x 5 x 1 x 2 x 1 : x 1 x 2 Your Turn 1. Calculate: a) x x 2 1 x 1 x 2 x b) 1 x 2 4 1 x 2 11
c) x 1 x 1 x x 1 d) x 2 x 2x x 2 e) x x 2 x 4 x 1 f) x 2 3 x 1 2x 2 x 3 g) x 3 x 4 : x2 9 x 2 h) x 2 x 1 : x x 2 2x 1 12
Keywords: monomial = monomio binomial = binomio trinomial = trinomio Polynomial = polinomio variable = variable constant = constante the unknown = la incógnita degree = grado term = término constant term = término independiente Numerical value of a polynomial = valor numérico de un un polinomio to plug in numbers for the variable = sustituir por números la variable To evaluate when x = = calcular el valor cuando x = like terms = términos semejantes unlike terms = términos no semejantes dividend = dividendo divisor = divisor quotient = cociente remainder = resto Ruffini's Rule = Regla de Ruffini Remainder Theorem = Teorema del Resto Factor Theorem = Teorema del Factor Root or zeroes of a polynomial = raíces o ceros de un polinomio to factorise = factorizar common factor = factor común common denominator = común denominador to put fractions over a common denominator = escribir las facciones con denominador común to cross-multiply = multiplicar en cruz Algebraic Fraction = Facción Algebraica 13