QA #7 Factors and Multiples

Questions and Answers Sheet 7 Factors and Multiples Question #1 Perform the indicated operation, and simplify if possible. Assume any factors you cancel are not zero. 4 5 + 𝑥 𝑥2 2 9 − 𝑥2 𝑥 =? Answer: 4 5 + 𝑥 𝑥2 2 9 − 𝑥2 𝑥 = 4 5 x2 ( + 2 ) x x = 4x + 5 9 2 2 x 2 − ) 2 − 9x x x Question #2 Write the expression as a sum and/or difference of logarithms, express powers as factors: 5𝑥(3𝑥 + 3)/𝑥 2 Answer: Step 1 Consider the logarithm log 5𝑥(3𝑥+3) 𝑥2 . Express the logarithm as sum and or difference of logarithm as follows. log 5𝑥(3𝑥+3) = log 𝑥2 5(3𝑥+3) 𝑥 𝑚 = log 5 (3𝑥 + 3) − log 𝑥 (∵ log 𝑏 (𝑚) − log𝑏 (𝑛) = log𝑏 ( )) 𝑛 = log 5 + log(3𝑥 + 3) − log 𝑥 (∵ log 𝑏 (𝑚) + log 𝑏 (𝑛) = log 𝑏 (𝑚𝑛)) Step 2 Answer: The logarithm as sum and or difference of logarithm is log 5 + log(3𝑥 + 3) − log 𝑥. Question #3 What are the linear and quadratic factors of the expression 8𝑥 4 − 8𝑥 Answer: Given: 8𝑥 4 − 8𝑥 8𝑥(𝑥 3 − 1) 𝑥 3 − 1 = (𝑥 − 1)(𝑥 2 + 𝑥 + 1) = 8𝑥(𝑥 − 1)(𝑥 2 + 𝑥 + 1) Step 2 Linear factor = 𝑥, (𝑥 − 1) quadratic factor = (𝑥 2 + 𝑥 + 1) Question #4 Which property does the following equality represent? 12𝑥𝑥14 = 14𝑥𝑥12 Answer: 12𝑥𝑥14 = 14𝑥𝑥12 The equality represents the Commutative Property of Multiplication, which states: 𝑎𝑥𝑥𝑏 = 𝑏𝑥𝑥𝑎 Question #5 A vihacle valued at $35,795 originally, depreciates at a rate of 17%/year. The equation 𝑦 = 35795(0.83)𝑥 represents the vehicle's value. x is time passed since purchase (in years) and y is the woth (in dollars). How much will the car be worth ten years after the purchase? Answer: We're given: 𝑦 = 35795(0.83)𝑥 𝑥 = 10 To find y, substitute x, 𝑦 = 35795(0.83)1 0 ≈ 35,795(0.1551604) ≈ $5553.97 Question #6 Find the greatest common factor in the following expressions: 10𝑥 3 𝑦, 12𝑥𝑦 2 Answer: The common factors are: 10𝑥 3 𝑦, 12𝑥𝑦 2 60𝑥 3 𝑦 2 (the least common multiplier) Question #7 Find the greatest common factor in the following expressions: 8𝑥 2 𝑏3 , 10𝑥 2 𝑏 Answer: The common factors are: 8𝑥 2 𝑏3 , 10𝑥 2 𝑏 40𝑥 2 𝑦 3 (the least common multiplier) Question #8 Find the greatest common factor in the following expressions: 28𝑥 2 𝑦 4 , 42𝑥 4 𝑦 4 Answer: The common factors are: 28𝑥 2 𝑦 4 , 42𝑥 4 𝑦 4 84𝑥 4 𝑦 4 (the least common multiplier) Question #9 Find the greatest common factor in the following expressions: 10𝑥 3 , 12𝑥 2 , 14𝑥 Answer: The common factors are: 10𝑥 3 , 12𝑥 2 , 14𝑥 2x (the greatest common divider) Question #10 Find the greatest common factor in the following expressions: 35𝑥 3 𝑦 2 , 10𝑥 4 𝑦, 5𝑥 5 𝑦 3 Answer: The common factors are: 35𝑥 3 𝑦 2 , 10𝑥 4 𝑦, 5𝑥 5 𝑦 3 70𝑥 5 𝑦 3 (the least common multiplier) Question #11 Find the greatest common factor in the following expressions: 27𝑥 2 𝑦 3 , 45𝑥 3 𝑦 4 Answer: The common factors are: 27𝑥 2 𝑦 3 , 45𝑥 3 𝑦 4 135𝑥 3 𝑦 4 (the least common multiplier) Question #12 Find the outcome. (3 + 2𝑥𝑥5 − 4)/(8/4 + 1) Answer: Given: (3 + 2𝑥𝑥5 − 4)/(8/4 + 1) Multiplication and division first: (3 + 2𝑥𝑥5 − 4)/(8/4 + 1) = (3 + 10 − 4)/(2 + 1) Substract and add to find the numerator and the denominator: (3 + 10 − 4)/(2 + 1) = 9/3 Divide: 9/3 = 3 The answer: (3 + 2𝑥𝑥5 − 4)/(8/4 + 1) = 3 Question #13 Simplify the following expression: (5/𝑡)4 Answer: Recall that (𝑎/𝑏)𝑛 = 𝑎𝑛 𝑥𝑥𝑏(−𝑛) In our case 𝑛 = 4, 𝑎 = 5 and 𝑏 = 𝑡. Substitute: (5/𝑡)4 = 54 𝑥𝑥𝑡 (−4) = 625𝑡 (−4) Question #14 Use inductive reasoning to predict the next number in 1,4,9,16,25,36,49, ? Answer: The next number is 64. Those are the natural number in the power of 2. e.g 12, 22, 32 …. Question #15 Use inductive reasoning to predict the next number in 1,5,12,22,35, ? Answer: The next nubmer is 51. Starting with 4 and 1. Each next turn we add 3 more. e.g. 1 + 4 = 5,5 + (4 + 3) = 12,12 + (4 + 3 + 3) = 22 … Question #16 Use inductive reasoning to predict the next number in 1/2,2/3,3/4,4/5,5/6,6/7, ? Answer: The next number is 7/8. Just next natural nubmer for both parts of the fraction. Question #17 Use inductive reasoning to predict the next number in 3,5,9,15,23,33, ? Answer: The next number is 45. Starting with 3 and 2. Each next turn we add 2 more. e.g. 3 + 2 = 5,5 + (2 + 2) = 9,9 + (2 + 2 + 2) = 15 … Question #18 Find the angle berween the two vectors, rounded to the nearest tenth of a degree: 𝑢 = 𝑖 + 𝑗 + 𝑘, 𝑣 = 3𝑠 − 4𝑘 Answer: Step 1 Given Data: Two vectors: 𝑢 = 𝑖 + 𝑗 + 𝑘 = (1,1,1) 𝑣 = 3𝑖 − 4𝑘 = (3,0, −4) The dot product of two vectors is, 𝑢 ⋅ 𝑣 = (1,1,1) ⋅ (3,0, −4) = (1)(3) + (1)(0) + (1)(−4) =3+0−4 =3−4 = −1 Step 2 The value of vector u is, ||𝑢|| = √(1)2 + (1)2 + (1)2 = √1 + 1 + 1 = √3 The value of vector v is, ||𝑢|| = √(3)2 + (0)2 + (−4)2 = √9 + 0 + 16 = √25 =5 Step 3 The angle between two vectors is, 𝑢⋅𝑣 𝑐𝑜𝑠𝜃 = ||𝑢||||𝑣|| Substitute the values in the above equation. cos 𝜃 = ( −1 √3)(5) 𝜃 = cos −1 (−0.1155) 𝜃 = 96.6∘ Thus, the angle between two vectors is 96.6∘ Question #19 Find the least common multiple of 𝑥 3 − 𝑥 2 + 𝑥 − 1𝑎𝑛𝑑𝑥 2 − 1 . Write the answer in factored form. Answer: Factor the given polynomials: 𝑥 3 − 𝑥 2 + 𝑥 − 1 = 𝑥 2 (𝑥 − 1 ) + (𝑥 − 1) = (𝑥 − 1)(𝑥 2 + 1) 𝑥 2 − 1 = (𝑥 − 1)(𝑥 + 1) The lowest common multiple is the product of all the factors of each polynomial to the highest degree. Each of the factors x-1, x+1, and 𝑥 2 + 1 all have a degree of 1 so the lowest common multiple is (𝑥 − 1)(𝑥 + 1)(𝑥 2 + 1) . Result: (𝑥 − 1)(𝑥 + 1)(𝑥 2 + 1) Question #20 How to factor 𝑥 2 − 2𝑥 Answer: Yes, the solution is x(x-2). The reason is that in both 𝑥 2 and 2x there is a common factor of x (x is in both...). You're dividing x and factoring it out to the front. So x is what's taken from both monomials (each piece of the expression) in this binomial.