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Chapter 3 Algebraic Fractions

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Chapter 3 Algebraic Fractions 5. Perfect Square Trinomial A trinomial is a perfect square if the middle term is twice the product of the square roots of the first and the third term. Some expression consists of three terms; two terms that are perfect squares, and third term which is twice the product of the square roots of the two perfect square terms. The two perfect square trinomials are positive and negative. x2 – 2xy + y2 = ( x – y )2 or (x – y) (x – y) Perfect Square Square of 1st 2nd Trinomial the Difference factor factor (Negative) of the Square Root x2 + 2xy + y2 = ( x + y)2 or (x + y) (x + y) Perfect Square Square of 1st 2nd Trinomial the Difference factor factor (Positive) of the Square Root Examples: a – 2ab + b2 = (a + b) 2 4x2 + 36x + 81 = (2x + 9)2 x2 – 10x + 25 = (x – 5)2 CHAPTER 3 ALGEBRAIC FRACTIONS Objectives 1. Define and give examples of basic algebraic fractions 2. Evaluate algebraic fractions Rational expression simple fractions are a quotient of two algebraic expressions or polynomials. Types of fractions: 1. Proper fraction is one whose degree of polynomial in the numerator is less than the degree of the polynomial in the denominator. Examples: 2 ; x + 1 x + 5 x2 – 2x + 3 2. Improper fraction is one whose degree of polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. Examples: x2 – 1 ; x – 1 x + 1 x + 1 3. Equivalent Fraction is a fraction which has the same value but different in the form. Examples: 16 = 8 ; ac = a 36 18 bc b 3.1 Reduction of Rational Expressions to lowest terms Objective Reduce a rational expression to its lowest term. It is easy to assess the value of a fraction when it is reduced to lowest terms. This is the main reason why we always reduce a fraction to lowest terms. An algebraic fraction is said to be in lowest terms if the numerator and the denominator do not have a common factor except 1. Procedure: 1. Determine the largest common factor of both numerator and denominator. 2. If the numerator or denominator is a polynomial, factor if necessary. 3. Divide the numerator and denominator by all common factors. Examples: 4xy = 2y ; x2 – 4 = (x + 2) (x – 2) = x + 2 10xz 5z 2x-4 2(x – 2) 2 As shown in the examples, knowledge of the different techniques of factoring polynomials is needed to be able to reduce an algebraic fraction to lowest terms. 3.2 Addition and Subtraction of Algebraic Fraction Objective Add and subtract algebraic fractions Addition and subtraction of fractions depend on the kinds of denominator of the fraction have. Addition and Subtraction of fraction having a common denominator To add or subtract fractions having a common denominator, add or subtract the numerators and place the result over the common denominator. Examples: 3x + 2x = 5x 4y 4y 4y 3.3 Multiplication of fraction Objective Multiply an algebraic fraction. To multiply fractions, multiply their numerators to obtain the numerator of the product, and multiply the denominators to obtain the denominator of the product. Factors common to numerator and denominator should, of course, be divided out. Example: ac . a2 = a3c bd b2 bd3 3.4 Division of fraction Objective Divide an algebraic fraction. To find the quotient of two fractions, multiply the numerator by the reciprocal of the denominator. As with multiplication, the numerator and denominator of each fraction should be put in factored form, and the result reduced to lowest term. Examples: 1. 12x2 + 7xy – 10y2) ÷ 9x2 – 4y2 x2 Solution: 2 2 2 2 12x + 7xy – 10y ) ÷ 9x – 4y 2 x 2 2 2 = (12x + 7xy – 10y ) . x 2 2 9x - 4y = (3x – 2y) (4x + 5y) x2 (3x – 2y) (3x + 2y) = (4x + 5y) x2 (3x + 2y) 2. x2(xy – y) . x2 – y4 ÷ x6 – x5y2 3x2 + 6xy x + y2 2x + 4y Solution: x2(x – y) . x2 – y4 ÷ x6 – x5y2 3x2 + 6xy x + y2 2x + 4y = xy(x – 1) . (x –y2) (x + y2) . 2 (x + 2y) 3x(x + 2y) x + y2 x5 (x – y2) = 2y (x2 – 1) 3x4 CHAPTER 4 EXPONENTS AND RADICALS 4.1 Exponent Objective Apply the laws of exponents in simplifying algebraic expressions. Definition of Terms . Exponent is the number written on the top right side of a quantity to indicate how many times that the quantity is used as a factor in multiplication. Thus, a4, 4 is the exponent. Factor is any one of two or more quantities that are multiplied. Thus, in axb = ab, a and b are factors. Power of a quantity is the product obtained when a quantity is multiplied by itself one or more times. Base of a power is the quantity that is multiplied by it one or more times. Thus, in 52 5 is the base, 2 is the exponent; the product of 5 x 5 = 25 is the power. 52 are read “5 squared” or “5 raised to the second power”. Square root of a quantity is one of its two equal factors. The quantity of 36 has two square roots +6 and -6. The position square quantity or number has two square roots. Laws of Exponents 1. The Multiplication Law To multiply two expressions having the same base, retain the base and add the exponents. That is am. an = am+n Examples: 3 .34= 31+4 = 35 42.4 = 42+1 = 43 2. The Division Law The quotient of a power with the same base is equal to the base raised to an exponent equal to the difference of the exponents of the numerator and denominator, that is: Examples: a. m/an = a m-n if m > n 45/43 = 45-3 = 42 = 16 b. an/an = 1 if m < n an-an c. an/an = a0 = 1 if m = n Any quantity (Except 0) with the zero exponents has a numerical value of 1. Example 50 = 1 3. The Power of a Product If a product is raised to a power, the result equals the product of each factor raised to that power that is (a . b) an . bn Example: (3. 2)4 = 34 . 24 = 81 . 16 = 1296 4. The Power of a Quotient The power of a quotient is equal to the quotient of their powers, that is (a/b)n = an/bn Example: (2/3)4 = 24/34 = 16 81 5. The Power of a Power If an expression containing a power is raised to a power retain the based and multiply the powers, that is (am)n = amn Example: [(-3)3]2 = (-3)3.2 = -36 = 729 6. Rational Exponent The fractional exponent means that the base is to be raised to a power indicated by the number of the fraction and the root is to be extracted whose index is the denominator of fractions. That is am/n = n am Examples: 8 2/3 = √ . √ = 4 2.53/4 = √ . 53 = 125 7. Negative Exponent Any quantity with a negative exponent is equal to the reciprocal of the quantity with the corresponding positive exponent. That is, a-m = 1/am Example: 3-3 = 1 = 1 32 9 .
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