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

2024
Views:

479

Pages:

2
Author:

nesikhiyawl0

Question 1 Find the HCF of 72 and 48 using the prime factorization method Answer Step 1 Given, 72 and 48 We have to find the HCF of 72 and 48 using the prime factorization method. Step 2 Doing the prime factorization of 72 and 48 72 = 2³ x 3² 48 = 2⁴ x 3¹ ⇒ HCF(72, 48) = 2³ x 3¹ (taken the lowest power of prime factors) ⇒ HCF(72, 48) = 24 Question 2 Give the prime factorization of the number. Use exponential form. 350 Answer Step 1 Consider the given number 350. Step 2 Obtain the prime factorization of the number 350 as follows. 350 = 10 × 35 = 2 × 5 × 5 × 7 [35 = 5 x 7 and 10 = 2 × 5] = 2 x (5 x 5) x 7 = 2 x 5² x 7 Thus, the prime factorization of the number 350 is 2 x 5² x 7. Question 3 Use the unique factorization theorem to write the following integers in standard factored form. 5733 Answer Step 1 According to unique factorisation theorem every number can be represented in the form of positive prime number with exponents. Various prime numbers are 2,3,5,7,9,11,.. Step 2 Divide the number with prime number, starting from least to determine whether it is a factor or not. 5733 = 3 × 1911 = 3² x 637 = 3² × 7 × 91 = 3² x 7² x 13 So, prime factorization of 5733 is 3² x 7² x 13. Question 4 Write the linear factorization and list all the zeros of p(x) = x⁴+x³- 8x² – 2x + 12. Answer Step 1 p(x) = x⁴+x³- 8x² – 2x + 12 ⇒ p(x) = x³(x - 2) + 3x²(x - 2) - 2x(x-2) -6(x - 2) ⇒ p(x) = (x - 2)(x³ + 3x² - 2x - 6) ⇒ p(x) = (x - 2)[x²(x+3)-2(x+3)] ⇒ p(x) = (x - 2)(x + 3)(x² - 2) ⇒p(x) = (x - 2)(x+3)(x+√2)(x - √2) Step 2 linear factorization of p(x). p(x) = (x - 2)(x + 3)(x + √2)(x - √2) Zeros of p(x): 2,-3, √2,-√2. Question 5 Factor the trinomial shown below. Show that your factorization is correct by multiplying the two binomial factors. 5x² + 13x-6 Answer Step 1 Given equation, 5x² + 13x - 6 First we find the factor. 5x² + 15x - 2x - 6 5x(x + 3) - 2(x - 3) (5x - 2)(x + 3) ...factor. Step 2 Now, Multiple factor (5x - 2)(x + 3) 5x(x + 3) - 2(x + 3) 5x² + 15x - 2x - 6 5x² + 13x - 6 Hence proved.