Lecture 2 - Algebraic Terms and Rational Expressions PDF

Summary

Algebraic lecture notes on algebraic terms and rational expressions, including distributive properties, FOIL method, factorisation, and special products. The lecture outlines various methods and examples for simplifying rational expressions, performing operations (multiplication and addition), and solving problems involving exponents and radicals.

Full Transcript

LECTURE 2 ALGEBRAIC TERMS AND RATIONAL EXPRESSIONS THE DISTRIBUTIVE PROPERTY A term is a quantity separated by a plus or minus sign. Example: 13x2y + 2xy +22xy2 + 15 The number in a term is called coefficient. A term without a variable is called constant....

LECTURE 2 ALGEBRAIC TERMS AND RATIONAL EXPRESSIONS THE DISTRIBUTIVE PROPERTY A term is a quantity separated by a plus or minus sign. Example: 13x2y + 2xy +22xy2 + 15 The number in a term is called coefficient. A term without a variable is called constant. 2 THE DISTRIBUTIVE PROPERTY Two terms are alike if they have the same variables and we can combine like terms by adding/subtracting coefficients on terms that are alike. Combine like terms: Example: 13x2 + 2x -3y + 2xy2 – 5x +5y – 8x2 = (13x2 - 8x2) + 2xy2 + (2x – 5x) + (5y -3y) = 5x2 + 2xy2 – 3x + 2y 3 THE DISTRIBUTIVE PROPERTY Use the Distributive Property in expanding expressions. Example 1: 10x (5y + 2xy -3x) = 10x(5y) + 10x(2xy) – 10x(3x) = 50xy + 20x2y – 30x2 Example 2: 3(2x2 – 5x + 2y) – 4x(3x -4) = 6x2 – 15x + 6y – 12x2 + 16x = -6x2 + x + 6y Guide: ( a + b + c)(d + e) = ad + bd + cd + ae + be + ce 4 THE DISTRIBUTIVE PROPERTY FOIL METHOD First x First + Outer x Outer + Inside x Inside + Last x Last 5 THE DISTRIBUTIVE PROPERTY FOIL METHOD First x First + Outer x Outer + Inside x Inside + Last x Last EXAMPLE 1: Expand the Expression (2x + 3)(x – 4) F O I L (2x+3)(x – 4) = 2x(x) + 2x(-4) + 3(x) + 3(-4) = 2x2 – 8x + 3x – 12 = 2x2 – 5x -12 6 THE DISTRIBUTIVE PROPERTY FOIL METHOD First x First + Outer x Outer + Inside x Inside + Last x Last EXAMPLE 2: Expand the Expression (3x + 5)(4x – 4) F O I L (3x+5)(4x – 4) = 3x(4x) + 3x(-4) + 5(4x) + 5(-4) = 12x2 – 12x + 20x – 20 = 12x2 + 8x -20 7 FACTORIZATION Factoring is the process of using the distributive Property in reverse. EXAMPLE: 6x2 – 12x + 15xy = 3x (2x – 4 + 5y) We identify what is common to each term then factor them out. 8 FACTORIZATION Many expressions having three terms can be factored so that the FOIL method gives us the original expression. EXAMPLE 1: Factor the expression x2 -4x + 3 x2 - 4x + 3 = (x )(x ) factors either 1,3 or -1, -3 = (x – 1)(x – 3) 9 FACTORIZATION Many expressions having three terms can be factored so that the FOIL method gives us the original expression. EXAMPLE 2: Factor the expression 4x2 - 4x - 15 4x2 – 4x – 15 = Factors of 4x2, (x, 4x), (2x, 2x) Factors of -15, (1,-15),(-1, 15), (-3, 5), (3, -5) = (2x + 3)(2x – 5) 10 FACTORIZATION The difference of two squares can be factored with the formula a2 – b2 = (a – b)(a +b) EXAMPLE 3: Factor the expression x2 - 9 x2 – 9 = (x – 3)(x + 3) EXAMPLE 4: Factor the expression 4x2 - 16 4x2 – 16 = (2x – 4)(2x + 4) 11 FACTORIZATION Special Products and Factoring 1. (x + y)(x – y) = x2 - y2 2. (x + y)2 = x2 + 2xy + y2 Example: (x + 5)2 = x2 + 2x(5) + 52 = x2 + 10x + 25 3. (x - y)2 = x2 - 2xy + y2 Example: (x - 4)2 = x2 - 2x(4) + 42 = x2 - 8x + 16 4. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz 5. (x3 + y3) = (x + y)(x2 - xy + y2) 6. (x3 - y3) = (x - y)(x2 + xy + y2) 12 RATIONAL EXPRESSIONS A rational expression (or any kind of fraction) is in lowest terms if the numerator and denominator have no common factors (other than 1). We simplify (or reduce) a fraction by dividing the numerator and denominator by their common factor(s). This process is also called cancelling. 𝑥 4𝑥−9 EXAMPLE: , 𝑥−4 2𝑦−𝑥−𝑧 13 RATIONAL EXPRESSIONS BASIC FRACTION OPERATION SUMMARY 1. Multiplying Fractions 𝑎 𝑐 𝑎𝑐 × = 𝑏 𝑑 𝑏𝑑 2. Dividing fractions 𝑎 𝑐 𝑎 𝑑 ÷ = 𝑥 𝑏 𝑑 𝑏 𝑐 3. Adding fractions (with same denominators) 𝑎 𝑐 𝑎+𝑐 + = 𝑏 𝑏 𝑏 4. Adding fractions (with different denominators) 𝑎 𝑐 𝑎𝑑+𝑏𝑐 + = 𝑏 𝑑 𝑏𝑑 5. Simplifying Fractions 𝑎𝑏𝑐 𝑎 = 𝑐𝑏𝑑 𝑑 14 RATIONAL EXPRESSIONS Simplify the rational expressions. 6x2y x2+ 2x −8 Example 1: Example 2: 2 15𝑥𝑦 x − 3x + 2 6x2y (3𝑥𝑦)(2𝑥) = (𝑥+4)(𝑥−2) = (𝑥−1)(𝑥−2) 15𝑥𝑦 (3𝑥𝑦)(5) 𝟐𝒙 𝒙+𝟒 = = 𝟓 𝒙−𝟏 15 RATIONAL EXPRESSIONS Find the product. 15xy2 8 Example 3: x 14 3𝑥𝑦 3𝑥𝑦 5𝑦.(4)(2) = 2 7.(3𝑥𝑦) 𝟐𝟎𝒚 = 𝟕 16 RATIONAL EXPRESSIONS Example 4: Perform the operations in the expression. 1 2 𝑥 + − 2𝑥 3𝑥 6 LCD of 2x, 3x, 6 = 6x 1 3 + 2 2 −(𝑥)(𝑥) = (6𝑥) 3 + 4 −x2 = 6𝑥 −x2+𝟕 = 𝟔𝒙 17 LAWS OF EXPONENTS (INDEX LAW) An exponent refers to the number of times a number is multiplied by itself 1. an = a x a x a x a … (n factors) 2. am x an = am + n Example: 23 x 24 = 23+4 = 27 = 128 xz (xz+2) = xz + (z+2) = x2z+2 am 3. an = am-n 55 Example: = 5 5-3 = 52 = 25 53x−5 x = x x-5 – (2x-2) = x–x-3 x2x −2 18 LAWS OF EXPONENTS (INDEX LAW) 4. (am)n = amn Example: (22)4 = 22 x 4 = 28 = 2256 𝑥 +2x (x ) = x x x+2 x(x+2) =x 5. (abc)m = ambmcm Example: (2xy)3 = 23x3y3 = 8x3y3 𝒂 𝒏 an 6. = 𝒃 bn 2𝑥 2 22x2 4x2 Example: = = 2 3𝑦 3 y 9y 2 2 19 LAWS OF EXPONENTS (INDEX LAW) 𝒏 7. am/n = 𝒂𝒎 Example: x(5+x)/x = 𝑥 x(5+x) 1 1 8. a-m = m and −m = am a a x Example: xy = 2 -2 y 2x = 2xy 2 y−2 9. a0 = 1 10. If am = an then m = n (provided a ≠ 0) 20 PROPERTIES OF RADICALS Radical - The √ symbol that is used to denote square root or nth roots. 𝒏 1. a1/n = 𝒂 2𝑦 Example: x1/2y = 𝑥 𝒏 2. am/n = 𝒂𝒎 Example: xxy/5 = 5 xxy 21 PROPERTIES OF RADICALS 𝒏 3. 𝒂 n =a Example: 5 𝑥 5 =x 𝒏 𝒏 𝒏 4. 𝒂 𝐱 𝒃 = 𝒂𝒃 3 3 𝑛 Example: x2 x 2𝑦 = 2x2𝑦 𝒏 𝒂 𝒏 𝒂 5. 𝒏 = 𝒃 𝒃 𝟒 𝟐𝟒𝒙𝒚 𝟒 𝟐𝟒𝒙𝒚 𝟒 𝟖𝒙𝒚 Example:. 𝟒 = = 𝟑𝒙𝒛 𝟑𝒙𝒛 𝒛 22 OTHER IMPORTANT PROPERTIES OF ALGEBRA 1. a x 0 = 0 2. If a x b = 0, then a = 0 or b = 0 or both a and b is equal to 0. 0 3. = 0, if a ≠ 0 𝑎 𝑎 4. = undefined 0 𝑎 5. =0 ∞ 23 EXPONENTS & RADICALS EXAMPLE PROBLEM 1 Rewrite the expression 3(2x3)4 Solution: 3(2x3)4 = 3(24 x3(4)) = 3 (16) x12 = 48 x12 24 EXPONENTS & RADICALS EXAMPLE PROBLEM 2 Solve for a in the equation a = 64x 4y Solution: a = 64x 4y a = (43)x 4y a = (43x) 4y a = 43x + y 25 EXPONENTS & RADICALS EXAMPLE PROBLEM 3 (ECE BOARD 1993) Find x from the equations: 27x = 9y 81y 3-x = 243 Solution: 81y 3-x = 243 27x = 9y 34y3-x = 35 33x = 32y , If am = an then m = n (provided a ≠ 0) – index law 34(3x/2)3-x = 35 so 3x = 2y and y = 3x/2 36x3-x = 35 36x-x = 35 35x = 35 5x = 5 x=1 26 EXPONENTS & RADICALS EXAMPLE PROBLEM 4 xy−1 4 x2y−2 3 Simplify: −2 3 ÷ −3 3 x y x y x4y−4 x−9y9 𝑥 6 −6 = x4-9-(-8)-6 y-4+9-12-(-6) Solution: x y −8 12 xy xy−1 4 x2y−2 3 = x-3 y-1 ÷ x y −2 3 x−3y3 𝟏 xy−1 4 x−3y3 3 = x−2y3 𝑥 x2y−2 x3 y x4y−4 x−9y9 𝑥 x y −8 12 x6y−6 27 EXPONENTS & RADICALS EXAMPLE PROBLEM 5 Simplify the following: 7a+2 – 8(7a+1) + 5(7a) + 49(7a-2) Solution: 7a+2 – 8(7a+1) + 5(7a) + 49(7a-2) am x an = am + n – index law = 7a72 – 8(7a71) + 5(7a) + 49(7a7-2) = 7a [72 – 8(7) + 5 + 72(7-2)] = 7a [49 – 56 +5 +1] = 7a (-1) = -7a 28 EXPONENTS & RADICALS EXAMPLE PROBLEM 6 Simplify the following: 100x2y4 Solution: 100x2y4 = 102x2(y2)2 = 10xy2 29 EXPONENTS & RADICALS EXAMPLE PROBLEM 7 12+ 80 Simplify the following: 4 Solution: 12+ 80 12+ 16(5) = 4 4 12+ 4 5 = 4 4 3+ 5 = 4 =3 + 5 30 EXPONENTS & RADICALS EXAMPLE PROBLEM 7 3 3 3 Simplify : 2x4 − 16x4 + 2 54x4 Solution: 3 3 3 = 2x4 − 16x4 + 2 54x4 3 3 = 2x4 − (8)(2x4) + 2 3 (27)(2x4) 3 3 3 = 2x4 −2 2x4 + 2(3) 2x4 3 = 2x4 [1 – 2 + (2)(3)] 3 = 𝟓 2x4 31 EXPONENTS & RADICALS EXAMPLE PROBLEM 8 (ME BOARD 1998) Find the value of x that will satisfy the following expression: 𝑥−2= 𝑥 +2 Solution: 𝑥−2= 𝑥 + 2 - square both sides 𝑥 − 2 2 = ( 𝑥 + 2)2 x -2 = x + 2(2)( 𝑥) + 4 x –x - 4 𝑥 = 4 + 2 - 4 𝑥 = 6 (square both sides) 16x = 36 𝟑𝟔 𝟗 x= = 𝟏𝟔 𝟒 32

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