Further Algebra Lecture 2 PDF

1 Further Algebra  Objectives 2 After completing this Lecture, you should be able to understand: Scientific notation Simplifying Rational Expressions Rational Expression operations Polynomials operations Recognize special products ...

1 Further Algebra  Objectives 2 After completing this Lecture, you should be able to understand: Scientific notation Simplifying Rational Expressions Rational Expression operations Polynomials operations Recognize special products  Scientific Notation A positive real number is written in scientific notation when it is written in the form 𝑎 × 10𝑛 , where 1 ≤ 𝑎 < 10 and 𝑛 is an integer. A reminder: Powers of 10 Power of 10−4 10−3 10−2 10−1 100 101 102 103 104 𝟏𝟎 Value 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 Warning: A number can be written in many ways using the powers of 10, but only one of them represents the scientific notation. For example, while 𝟏𝟔. 𝟓 = 𝟏𝟔𝟓 × 𝟏𝟎−𝟏 = 𝟏𝟔. 𝟓 × 𝟏𝟎𝟎 = 𝟏. 𝟔𝟓 × 𝟏𝟎𝟏 = 𝟎. 𝟏𝟔𝟓 × 𝟏𝟎𝟐 is true, only 𝟏. 𝟔𝟓 × 𝟏𝟎𝟏 is the scientific notation of 16.5  Example 5: Writing Numbers in Scientific Notation Write each number in scientific notation. a) 295000000 b) 0.00000000105 Solution: a) 295000000 = 2.95 × 108 b) 0.00000000105 = 1.05 × 10−9 Try & Check Write each number in scientific notation. a) 6400000 b) 0.000000002  Example 6: Converting from Scientific Notation to Decimals Rewrite the following numbers in decimal form. a) 2.05 × 104 b) 5.01 × 10−5 Solution: a) 2.05 × 104 = 2.05 × 10000 = 20500 b) 5.01 × 10−5 = 5.01 × 0.00001 = 0.0000501 Try & Check Rewrite the following numbers in decimal form. a) 1.05 × 106 b) 4. 1 × 10−6 Rational Expression A Rational Expression is a fraction where the numerator and denominator are polynomials Some Examples: 𝑥+1 𝑥𝑦 2𝑥 3𝑥+1 , , , 𝑥 𝑥 2 +𝑥𝑦+𝑦 2 𝑥 2 +1 𝑥 2 −4 Some Remarks on Rational Expressions: 1. The denominator of a rational expression cannot be 0. 2. When the denominator of a rational expression is 0, we say that the expression is undefined. 𝑥+1 For example, in order for 𝑥 not to be undefined, the variable 𝑥 must be different from 0 Same rule of fractions of the previous lectures will be used  Simplifying Rational Expression Theorem (The Cancellation Law) If 𝑎, 𝑏 and 𝑐 are real numbers such that 𝑏 ≠ 0 and 𝑐 ≠ 0, then 𝑎𝑐 𝑎 =. 𝑏𝑐 𝑏 Example : Simplifying Rational Expressions 𝑥 4 +𝑥 3 Simplify the rational expression 𝑥 3 +𝑥 2. Factor the numerator. First, factor out 𝑥4 + 𝑥3 𝑥3 𝑥 + 1 the GCF, 𝑥 3. 3 2 = 𝑥 +𝑥 𝑥3 + 𝑥2 Factor the denominator, First, factor out 𝑥3 𝑥 + 1 3−2 𝑥+1 = 2 =𝑥 the GCF, 𝑥 2. 𝑥 (𝑥 + 1ሻ 𝑥+1 Simplify. =𝑥  Example : Simplifying Rational Expression 25−𝑥 2 Simplify 𝑥 2 −7𝑥+10. 25 − 𝑥 2 Solution 𝑥 2 − 7𝑥 + 10 Factor the numerator (use the factoring of 5−𝑥 5+𝑥 = the difference of two squares). 𝑥 2 − 7𝑥 + 10 5−𝑥 5+𝑥 Factor the denominator. = 𝑥 − 5ሻ(𝑥 − 2 In the binomial 5 − 𝑥, factor out −1; 5 (−1ሻ 𝑥 − 5 5 + 𝑥 − 𝑥 = (−1ሻ(𝑥 − 5ሻ. = 𝑥 − 5ሻ(𝑥 − 2 Identify common factors (−1ሻ 𝑥 − 5 5 + 𝑥 = 𝑥 − 5ሻ (𝑥 − 2 Simplify 𝑥+5 =− 𝑥−2  Try & Check 4𝑥 3 +8𝑥 2 Simplify 𝑥 2 +2𝑥 Example 5: Simplifying Rational Expressions containing factors that are additive inverses Solution Reminder: 𝑎−𝑏 =− 𝑏−𝑎 = −1(𝑏 − 𝑎ሻ Try& Check Addition and Subtraction of Rational Expression Addition and Subtraction of Fractions with Common Denominator If 𝑎, 𝑏, 𝑐 are real numbers such that 𝑐 ≠ 0 then 𝑎 𝑏 𝑎+𝑏 𝑎 𝑏 𝑎−𝑏 + = and − =. 𝑐 𝑐 𝑐 𝑐 𝑐 𝑐 Example : Adding or Subtraction Rational Expression Perform the following operations and simplify 𝑥 2 +1 2𝑥 𝑥 2 +1 2 a. + b. − 𝑥+1 𝑥+1 𝑥+1 𝑥+1 a. 𝑥 2 +1 2𝑥 𝑥 2 +1+2𝑥 + = Add the numerators. 𝑥+1 𝑥+1 𝑥+1 𝑥 2 + 2𝑥 + 1 = 𝑥+1  continued Factor the numerator 𝑥+1 2 = 𝑥+1 Simplify. =𝑥+1 b. 𝑥2 + 1 2 𝑥2 + 1 − 2 Subtract the numerators. − = 𝑥+1 𝑥+1 𝑥+1 𝑥2 − 1 Simplify = 𝑥+1 Factor the numerator 𝑥 − 1ሻ(𝑥 + 1 = 𝑥+1 Simplify =𝑥−1  Multiplying and Dividing Rational Expression To multiply rational expressions, we always simplify each expression before multiplying. If 𝑎, 𝑏, 𝑐 and 𝑑 are real numbers such that 𝑏 ≠ 0, and 𝑑 𝑎 𝑐 𝑎𝑐 ≠ 0 then ⋅ =. 𝑏 𝑑 𝑏𝑑 Example 7: Multiplying Rational Expressions 3𝑥𝑦 𝑥 2 +6𝑥+9 Simplify the product: 𝑥 2 𝑦+3𝑥𝑦 ⋅ 3𝑥+9. 3𝑥𝑦 𝑥+3 2 Factor all numerators and all = ⋅ 𝑥𝑦(𝑥 + 3ሻ 3(𝑥 + 3ሻ denominators. 3𝑥𝑦 𝑥+3 2 Identify common factors. = ⋅ 𝑥𝑦(𝑥 + 3 3(𝑥 + 3ሻ ሻ Simplify. =1  Division of Fractions Example : Dividing Rational Expression 𝑝2 −4𝑝−12 2𝑝2 +6𝑝+4 Simplify: 𝑝2 −6𝑝 ÷ 8𝑝6. Solution: Rewrite the division as a multiplication by 𝑝2 − 4𝑝 − 12 8𝑝6 = ⋅ 2 the reciprocal. 𝑝2 − 6𝑝 2𝑝 + 6𝑝 + 4 𝑝−6 𝑝+2 8𝑝6 Factor all terms. = ⋅ 𝑝 𝑝−6 2(𝑝2 + 3𝑝 + 2ሻ 𝑝−6 𝑝+2 8𝑝6 Factor 𝑝2 + 3𝑝 + 2. = ⋅ 𝑝 𝑝−6 2 𝑝 + 2 (𝑝 + 1ሻ 𝑝−6 𝑝+2 2 ⋅ 4 ⋅ 𝑝 ⋅ 𝑝5 Find common factors. = ⋅ 𝑝 𝑝−6 2 𝑝 + 2 (𝑝 + 1ሻ 4𝑝5 Simplify = 𝑝+1  Try & Check Simplify: 𝑤 2 −3𝑤+5 𝑤 2 +10𝑤+16 ÷. 𝑤 2 −4 𝑤 2 −1  Polynomial 15 A monomial is a real number or a product of a real number and one or more variables with non negative integer exponents. The number is called the coefficient of the monomial. Example: 1 1 − 𝑥 2 𝑦𝑧 is a monomial in 𝑥 , 𝑦 𝑎𝑛𝑑 𝑧, its coefficient is −. 2 2 −𝑢2 𝑣 3 is a monomial in u 𝑎𝑛𝑑 𝑣, its coefficient is −1. 15 is a constant monomial, its coefficient is 15. The degree of a monomial is the sum of the exponents of its variables. 1 The degree of − 𝑥 2 𝑦𝑧 is 4. 2 The degree of −𝑢2 𝑣 3 is 5. The degree of 15 = 15𝑥0 is 0. The number 0 has no defined degree.  Polynomial 16 A monomial in one variable, 𝒂𝒙𝒌 , is the product of a constant and a variable raised to a nonnegative-integer power. The constant 𝒂 is called the coefficient of the monomial, and 𝒌 is called the degree of the monomial. A polynomial is the sum of monomials. The monomials that are part of a polynomial are called terms. Example: 1 1 − 𝑥 2 is a monomial in 𝑥, its coefficient is − and its degree is 2 2 2 A polynomial is a monomial or a sum of monomials. Each monomial in that sum is a term of the polynomial. A polynomial with two terms is a binomial, for example : 𝑥 3 + 4.7 A polynomial with three terms is a trinomial, for example : 𝑥 4 + 2𝑥 + 9 The degree of a polynomial is the degree of its term with the highest degree. Any non zero real number is a polynomial with degree 0. The number 0 is a polynomial with no defined degree, it is called the zero polynomial.  Polynomial 17 A polynomial in 𝒙 is an algebraic expression of the form 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 where 𝑎𝑛 , 𝑎𝑛−1 , … , 𝑎1 , 𝑎0 are real numbers, with 𝑎𝑛 ≠ 0, and 𝑛 is a nonnegative integer. The degree of the polynomial is 𝑛, the leading coefficient is 𝑎𝑛 , and the constant term is 𝑎0. Example: Write the polynomial 3𝑥 2 − 8 + 14𝑥 3 − 20𝑥 8 + 𝑥 in standard form and state its degree, leading coefficient, and constant term. Solution: Standard form: −20𝑥 8 + 14𝑥 3 + 3𝑥 2 + 𝑥 − 8 Degree: 8 Leading coefficient: −20 Constant term: −8  Add and Subtract polynomials 18 Polynomials are added and subtracted by combining like terms. Like terms are terms having the same variables and exponents. Like terms can be combined by adding their coefficients. 5𝑥 3 − 2𝑥 − 𝑥 3 − 2𝑥 3 = 5𝑥 3 − 𝑥 3 − 2𝑥 3 − 2𝑥 = 5𝑥 3 − 𝑥 3 − 2𝑥 3 − 2𝑥 = 2𝑥 3 − 2𝑥 Example: Find the difference and simplify 3𝑥 3 − 2𝑥 + 1 − 𝑥 2 + 5𝑥 − 9. 3𝑥 3 − 2𝑥 + 1 − 𝑥 2 + 5𝑥 − 9 = 3𝑥 3 − 2𝑥 + 1 − 𝑥 2 − 5𝑥 + 9 = 3𝑥 3 − 𝑥 2 − 2𝑥 − 5𝑥 + 9 + 1 =3𝑥 3 − 𝑥 2 − 7𝑥 + 10 CTECH-215 COLLEGE OF APPLIED MEDICAL SCIENCES  Multiply polynomials. 19 Multiplying a Monomial and a Polynomial Multiplying Two Polynomials Find the product and simplify Multiply and simplify 5𝑥 2 −𝑥 3 + 7𝑥 2𝑥 2 − 3𝑥 + 1 𝑥 2 − 5𝑥 + 7 2𝑥 2 + 1 𝑥 2 − 5𝑥 + 7 = 5𝑥 2 −𝑥 3 + 7𝑥 = −5𝑥 2 (𝑥 3 ሻ + 5𝑥 2 (7𝑥) = 2𝑥 2 𝑥 2 − 5𝑥 + 7 +1 𝑥 2 − 5𝑥 + 7 = 2𝑥 4 − 10𝑥 3 + 14𝑥 2 + 𝑥 2 − 5𝑥 + 7 = −5𝑥 5 + 35𝑥 3 = 2𝑥 4 − 10𝑥 3 + 15𝑥 2 − 5𝑥 + 7 CTECH-215 COLLEGE OF APPLIED MEDICAL SCIENCES  Special Products 20 𝑎 − 𝑏 𝑎 + 𝑏 = 𝑎2 − 𝑏 2 Difference of two Squares 𝑎+𝑏 2 = 𝑎 2 + 2𝑎𝑏 + 𝑏2 Square of a binomial sum 𝑎−𝑏 2 = 𝑎 2 − 2𝑎𝑏 + 𝑏2 Square of a binomial difference 𝑎3 − 𝑏3 = 𝑎 − 𝑏 𝑎2 + 𝑎𝑏 + 𝑏2 Difference of cubs 𝑎3 + 𝑏3 = 𝑎 + 𝑏 𝑎2 − 𝑎𝑏 + 𝑏2 Sum of cubs Example : Find the following: a) (3𝑥 + 1ሻ2 b) (3 − 𝑦ሻ2 (c) (4𝑎 + 2ሻ(4𝑎 − 2ሻ d) 64 − 8𝑥 3 3 (d) 27 + 125𝑦 CTECH-215 COLLEGE OF APPLIED MEDICAL SCIENCES DEPAENT OF CARD EXAMPLE Factoring a Perfect Square Factor k2 + 20k + 100.  k 2  20k  100   k  10  2 Check : 2  k 10  20k Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley  Example Factor each trinomial.   x  12  2 x  24 x  144 2   5 x  3 2 25 x  30 x  9 2 36a 2  20a  25 prime  2 x  9 x 2  42 x  49   2 x  3x  7  2 18 x  84 x  98 x 3 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example difference of two squares Determine whether the binomial is a difference of two squares. If so, factor. If not, explain. 5. 9x2 – 144y4 (3x + 12y2)(3x – 12y2) 6. 30x2 – 64y2 not a difference of two squares; 30x2 is not a perfect square 7. 121x2 – 4y8 (11x + 2y4)(11x – 2y4)  EXAMPLE Factoring Sums and difference of Cubes Factor each polynomial. 𝑝3 + 64 = 𝑝 + 4 𝑝2 − 4𝑝 + 16 27𝑥 3 − 64𝑦 3 = 3𝑥 + 4𝑦 9𝑥 2 + 12𝑥𝑦 + 16𝑦 2 512𝑎6 + 𝑏3 = 8𝑎2 + 𝑏 64𝑎 4 − 8𝑎2 𝑏 + 𝑏2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6.4 - 24 END OF LECTURE Week 2