Give an example of each of the following.a. A simple linear factor________________b. A repeated linear factor________________c. A simple irreducible quadratic factor________________d. A repealed irreducible quadratic factor
Problem 2ESolution:-Step1a. A simple linear factordefinition: linear factor 1) words: “linear factor with root r ” 2) usage: r is a some real number 3) meaning: the expression ( x r) Example The polynomial f(x)= can be written as a product of linear factors, f(x)=(x+1)(x-1)(x-2) . Step2b. A repeated linear factorExample 2x+2=Ax-A+bAt A=2-A+B=2-2+B=2B=2+2=4 Step3c. A simple irreducible quadratic factorExample I =First, complete the square: + 4x + 7 = + 3. Thus I = In the first of these two integrals, the numerator x is not a constant multiple of the derivative 2(x + 2) of the denominator, so substitution does not work. Substitute for the quantity in the completion of the square: let y = x+ 2, so x = y 2 and dx = dy. Hence I = Now split it up. I is the sum of =And The final answer, expressed in terms of x is ++ C. Step4d. A repeated irreducible quadratic factorExamplef (x) = Comparing numerators, we have5x2+20x+6 = A(x+1)2+Bx(x+1)+Cx.Taking x = 0, we find A = 6. Likewise, taking x = -1, we find that C = 9. To determine B, substitute any convenient value for x, say x = 1. (Unfortunately, notice that there is no choice of x that will make the two terms containing Aand C both zero, without also making the term containing B zero.) You should find that B = -1. So, we have = 6ln | x| -ln | x+1| -9(x+1)-1+c.