University:
North Central College
Course:
MATH 152 | Calculus II
Academic year:

2024
Views:

236

Pages:

3
Author:

nesikhiyawl0

Question 1 Factor completely: x² - x - 12 Answer Step 1 Factorization is the process of splitting expression in to factor form. After factorization it is written in to the form of product term which when multiplied give the original expression. For example x² + 5x + 6 = x² + 3x + 2x + 6 = x(x+3) + 2(x+3) = (x+3)(x+2) Step 2 The term x² - x - 12 is factorize as below Rewrite the -x term as -4x+3x Then take x common from x² - 4x term and 3 from 3x-12 term x² - x - 12 = x² - 4x + 3x - 12 = x(x-4) + 3(x-4) = (x-4)(x+3) Hence x² - x - 12 is factorize as (x-4)(x+3) Question 2 Factor each polynomial by removing the common monomial factor. ax - a Answer Step 1 Given: The polynomial ax - a, To determine: The factorization of the given polynomial. Step 2 Explanation: Factorizing the given polynomial, ax - a ⇒ a(x - 1) [Taking a as common factor from both terms] Answer: ax - a = a(x-1) Question 3 The polynomial y = x⁵ + x³ - x² - 1 factors into (x² + 1)(x³ - 1). What is the multiplicity of the root (x³ - 1)? a. 3 b. 2 c. 1 d. 5 Answer Step 1 Polynomial y = x⁵ + x³ - x² - 1 Factored form: (x² + 1)(x³ - 1) Equate (x³ - 1) = 0 +1 +1 x³ = 1 Taking cuberoot on both the sides, x=1 (Zero) According to the algebraic fundamental theorem, the multiplicity of a root is the number of times it appears in the polynomial's complete factorization. As the power of (x - 1) is 1. Therefore, its multiplicity is 1. Step 2 Ans: Multiplicity of the root (x³ - 1) is 1. So, the correct option is (c). Question 4 What is the prime factorization of 120? 2³.3.5 2.3⁶ 2².5 2³.7² Answer Step 1 Write all the numbers that are factors (divide) of 120. Step 2 Factors of 120 are: 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120 Out of these numbers only 2,3,5 are prime. Step 3 So multiplying 2,3,5 should give 120, 120 = 2³ x 3 x 5 should be the answer. Question 5 The prime factorization of 7007 is 7.11³.13 7³.11.13 7².11.13 71113 Answer Step 1 Prime factorization of 7007 7007 = 1001 7 1001 = 143 7 143 = 13 11 13 =1 13 7007=7,7,11,13 Step 2 Answer: 7007 = 7²·11·13