Polynomial factorization #24 - Questions and Answers

Question 1 How do you find the product of (a+5)(a-6)? Answer 1 Explanation: To solve this, we expand/distribute using FOIL: As you can see in this image, the first thing we do is multiply the firsts: aa = a² Then the outers: a-6=-6a Then the inners: 5a=5a and finally the lasts: 5-6=-30 Now combine them all together: a² - 6a + 5a - 30 We can still simplify the like terms. a² - 6a + 5a - 30 Combine like terms and get the final answer: a² - a - 30 Answer 2 (a+5)(a-6) a(a-6)+5(a-6) Notice that the + sign followed the 5 a² - 6a + 5a - 30 but -6a+5a is the same as -a giving. a² - a - 30 Question 2 How do you multiply (7x-6) (5x+6)? Answer Explanation: You can use the Distributive Law to get: (7x-6)(5x+6)=(7x5x)+(7x6)+(-65x)+(-66)= = 35x² + 42x – 30x - 36 Collect similar terms to get: = 35x² + 12x – 36 Question 3 Average: Fundamental Concepts and Qualities I'm going to start a new quantitative aptitude topic today called Average. It is a pretty straightforward subject that only requires basic mathematical computations. The average concept has several uses. In the following session, I will go over its uses. I'll start by trying to explain the fundamentals of this subject to you. Answer The term "average" simply refers to the mean value of all provided observations, or, alternatively, the arithmetic mean of observations. Average = (Sum of all observations)/ (Number of observations) Examplel: Calculate the average of the subsequent observations: 3, 4, 8, 12, 2, 5, 1 Solution: Average = (Sum of all observations)/ (Number of observations) Average = (3+ 4+ 8+ 12+ 2+ 5+ 1)/7 = 35/7 = 5 So, Average = 5 i) Between the maximum and least observation is the average. ii) The average will also be multiplied by the same value if the value of each observation is multiplied by some value N. i.e.N. For example: Assume the prior collection of findings. If all observations are multiplied by 2 then the new observations will be as follows.: 6, 8, 16, 24, 4, 10, 2 New Average = (70)/7 = 10 = 2(5) = 2 × Old Average iii) The average will change by the same amount whether the value of each observation is increased or decreased by some amount. For example: Continuing with the same example. If 2 is added to all observations, then new observations will be as follows: 5, 6, 10, 14, 4, 7, 3 New Average = (49)/7 = 7 = (5 + 2) = 2 + Old Average iv) Similarly, if each observation is divided by some number, then average will also be divided by same number. For example: If 2 is divided from all observations, then new observations will be as follows: 1.5, 2, 4, 6, 1, 2.5, 0.5 New Average = (17.5)/7 = 2.5 = 5/2 = Old Average/2 I can therefore state that any general procedure done to observations Example2: Find an average of first 20 natural numbers. Solution: Average = (Sum of first 20 natural numbers)/ (20) Now, we know that Sum of first n natural numbers = ((n)(n+1))/2 Therefore, Sum of first 20 natural numbers = (20×21)/2 Average = (20×21)/ (2 × 20) = 10.5 Example3: The second number in a set of three numbers is twice the first and three times the third. Find the greatest number if the average of these numbers is 44. Solution: Let x be the third number According to question, second number = 3x = 2(first number) Therefore, first number = (3x)/2 second number = 3x and third number = x Now, average = 44 = (x + 3x + (3x)/2)/3 ⇒ (11x)/2 = 44 × 3 x = 24 So, largest number i.e. (3x) = 72 Example4: Average of four consecutive even numbers is 27. Find the numbers. Solution: Let x, x+2, x+4 and x+6 be the four consecutive even numbers. According to question, ((x) + (x+2) + (x+4) + (x+6))/4 = 27 (4x + 120)/4 = 27 x = 24 Therefore, numbers are 24, 26, 28, 30 Special Case finding the average speed Suppose a man covers a certain distance at x km/hr and covers an equal distance at y km/hr. The average speed during the whole distance covered will be (2xy)/ (x+y) Example5: A bike covers certain distance from A to B at 50 km/hr speed and returns back to A at 56 km/hr. Find the average speed of the bike during the whole journey. A B 50km/hr 56km/hr Solution: Average speed = ((2xy)(x + y)) = (2x (50) × (56)) / (50+56) ⇒ 52.83 km/hr Answer 2 The average, also known as the arithmetic mean, is a fundamental concept in mathematics and statistics. It is used to summarize a set of values by calculating their mean value. The average is calculated by dividing the sum of all the values by the total number of values. Let's say we have a set of n values denoted as x₁, x₂, x₃, ..., xₙ. The average of these values, denoted by x (pronounced as 'x-bar'), is calculated using the following formula: x = 1/n Σ (i=1 to n) xᵢ In this formula, the sum of all the values, denoted by ∑ (i=1 to n) xᵢ, represents adding up all the values from x₁ to xₙ. The division by n at the beginning calculates the average. For example, let's consider a set of 5 values: 10, 15, 20, 25, and 30. We can calculate the average as follows: x = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20 So, the average of the given set of values is 20. The average has several qualities that make it useful in various applications. Some of the key qualities of averages include: Central Tendency: The average represents the central value or typical value of a set of data. It provides a representative value that summarizes the entire data set. Balance: The sum of deviations above the average is equal to the sum of deviations below the average. In other words, positive deviations cancel out negative deviations, resulting in a balanced measure. Affected by Outliers: The average is sensitive to outliers, which are extreme values that differ significantly from the other values in the data set. Outliers can distort the average value, making it less representative of the majority of the data. Simple Computations: Calculating the average is relatively simple and straightforward, requiring basic addition and division operations. It allows for easy comparisons and computations involving a set of values. These are some of the fundamental concepts and qualities of averages. Understanding and applying the average is essential in various quantitative aptitude problems, statistics, and everyday life scenarios involving data analysis. Answer 3 The concept of average, denoted as x, is a fundamental concept in mathematics. It is used to represent the central tendency or typical value of a set of numbers. To calculate the average, we add up all the values in the set and divide the sum by the total number of values. Let's consider a set of n numbers: x₁, x₂, x₃, ..., xₙ. The sum of these numbers is represented as Σ (i=1 to n) xᵢ. To find the average, we divide this sum by n: x = 1/n Σ (i=1 to n) xᵢ For example, if we have a set of three numbers: 4, 7, 9, we can calculate the average as follows: x = (4 + 7 + 9) / 3 = 20 / 3 So, the average of the set 4, 7, 9 is 20/3. The concept of average is widely used in various scenarios. It helps us understand the typical value of a data set, such as the average test score of a class or the average monthly temperature. It is also used in solving problems related to ratios and proportions. Understanding the concept of average is essential as it forms the basis for more advanced statistical calculations. By calculating the average, we can gain insights into the data and make informed decisions. In summary, the average is a measure of central tendency that represents the typical value of a set of numbers. It is calculated by dividing the sum of all the values by the total number of values in the set. Question 4 Write P into factored form P(x) = x³ + 3x² + 4x + 12 Answer Step 1 Definition used Factorization For writing cubic polynomial, We can try to find the greatest common factors from the first two terms and the last two terms. Now if we will get common factor from these terms, we will find and write the polynomial in factored form. Step 2 Given P(x) = x³ + 3x² + 4x + 12 = x²(x + 3) + 4(x+3) = (x + 3)(x²+4) Step 3 Result: P(x) in factored form is P(x) = (x + 3)(x²+4)