Questions and Answers #58 Polynomial Factorization

Questions and Answers Sheet 58 Polynomial Factorization Question #1 Consider the following polynomials over 𝑍8 where a is written for [a] in 𝑍8 : 𝑓(𝑥) = 2𝑥 3 + 7𝑥 + 4, 𝑔(𝑥) = 4𝑥 2 + 4𝑥 + 6, ℎ(𝑥) = 6𝑥 2 + 3. Find each of the following polynomials with all coefficients in 𝑍8 , 𝑔(𝑥) + ℎ(𝑥) Answer: Step 1 When adding two polynomials coefficients of like powers are added. With coefficients of polynomials in 𝑍8 again the same approach is taken to add the two polynomials. While adding it is to be noted that in 𝑍8 the coefficients can take values from 0,1,2,3,4,5,6,7. And after adding the coefficients resulting term is taken modulo 8. For example 7+8=15 but this is not in 𝑍8 so taking it modulo 8 gives 7, that is, 7+8=7 modulo 8. So in 𝑍8 , 7+8=7. Step 2 Given polynomials to be added are 𝑔(𝑥) = 4𝑥 2 + 4𝑥 + 6, ℎ(𝑥) = 6𝑥 2 + 3. To add them, add the coefficients of like terms and then apply modulo 8. 𝑔(𝑥) + ℎ(𝑥) = (4𝑥 2 + 4𝑥 + 6) + (6𝑥 2 + 3) = (4𝑥 2 + 6𝑥 2 ) + 4𝑥 + (6 + 3) = 10𝑥 2 + 4𝑥 + 9 = 2𝑥 2 + 4𝑥 + 1 modulo 8 Hence, 𝑔(𝑥) + ℎ(𝑥) = 2𝑥 2 + 4𝑥 + 1. Question #2 Give the degree of the polynomial. 2𝑣 5 𝑢3 𝑥 + 3𝑢10 − 6 − 𝑥 6 𝑣 5 Answer: Step 1 We have to find Step 2 Degree of the given polynomial is 6+5= 11 Ans: 11 Question #3 How do you express −16 + 5𝑓 8 − 7𝑓 3 in standard form? Answer: Standard form simply refers to the format of a mathematical expression where the terms are arranged by decreasing order of degree. Where the degree is determined by the exponent value of the variable of each term. For the example provided −16 + 5𝑓 8 − 7𝑓 3 The highest degree is 8 in the term 5𝑓 8 The next degree is 3 in the term −7𝑓 3 and the -16 term has a degree of zero because there is no variable. Therefore the standard form for this expression would be 5𝑓 8 − 7𝑓 3 − 16 Question #4 1 Is 𝑥 2 + 3𝑥 2 a polynomial? Answer: Explanation: 1 𝑥 2 + 3𝑥 2 = 𝑥 2 + 3√𝑥 Definition of a Polynomial: 1) 2 or more terms 2) No division by any variable 3) Exponents can only be natural numbers (0, 1, 2...) 4) Finite number of terms This expression fails rule #3 so it is not a polynomial. Question #5 2 How do you write 𝑦 = 𝑥 + 5 in standard form? 3 Answer: Explanation: Multiply by 3 to get rid of the fraction: 3y=2x+15 Move the x's to the left. Don't forget the sign change: -2x+3y=15 Now reverse all signs to make the x part positive: 2x-3y=-15 graph{2/3x+5 [-12.33, 7.67, -1.76, 8.24]} Question #6 How do you write 𝑥 4 − 9 + 3𝑥 2 − 𝑥 3 in standard form? Answer: Explanation: 𝑥 4 − 9 + 3𝑥 2 − 𝑥 3 In the standard form, you need to write the exponents in the decreasing order of the powers. ⇒ 𝑥 4 − 𝑥 3 + 3𝑥 2 − 9 Question #7 How do you write 𝑥 2 + 𝑥 4 − 6 in standard form? Answer: Explanation: When converting to Standard Form, simply arrange the values in order from greatest to smallest, with unknown values (x) being arranged first, then known values (6). 𝑥4 + 𝑥2 − 6 Question #8 How do you write 4 − 𝑥 + 2𝑥 2 in standard form? Answer: Explanation: Standard Form is in the form 𝐴𝑥 2 + 𝐵𝑥 + 𝐶 In this trinomial the A would be 2 The B would be -1 The C would be +4 2𝑥 2 − 𝑥 + 4 Question #9 How do you write 6 + 7𝑥 2 in standard form? Answer: Explanation: 6 + 7𝑥 2 7𝑥 2 + 6 is the standard form. Degree of polynomial - 2. No. of terms - 2. Question #10 What is the degree of the monomial 3𝑥 2 𝑦 3 ? Answer: The degree of a monomial is the sum of the degrees of all variables in the monomial expression. For 3𝑥 2 𝑦 3 the degree is 5(=2+3)