Questions and Answers #52 Polynomial Factorization

Questions and Answers Sheet 52 Polynomial Factorization Question #1 Find the degree of the polynomial. −2𝑥 3 + 4𝑥 2 − 8𝑥 + 10 Answer: Step 1 The given polynomial is −2𝑥 3 + 4𝑥 2 − 8𝑥 + 10 Step 2 It is known that, the degree of a polynomial is the highest power of the variable in a polynomial expression. consider the polynomial −2𝑥 3 + 4𝑥 2 − 8𝑥 + 10. Here −2𝑥 3 , 4𝑥 2 , −8𝑥, 10 are the terms where −2𝑥 3 is a leading term and 10 is a constant term. The coefficients of the polynomial are -2, 4 and -8. The degree of the polynomial −2𝑥 3 + 4𝑥 2 − 8𝑥 + 10 is 3. Question #2 Add the two polynomials. (−3𝑦 4 + 6𝑦 2 + 7) and (10𝑦 4 − 7𝑦 2 − 2𝑦 − 4) Answer: Step 1 We need to add the following two polynomials (−3𝑦 4 + 6𝑦 2 + 7) and (10𝑦 4 − 7𝑦 2 − 2𝑦 − 4) Step 2 The polynomials can be added and simplified as following (−3𝑦 4 + 6𝑦 2 + 7) + (10𝑦 4 − 7𝑦 2 − 2𝑦 − 4) = −3𝑦 4 + 6𝑦 2 + 7 + 10𝑦 4 − 7𝑦 2 − 2𝑦 − 4 = (−3𝑦 4 + 10𝑦 4 ) + (6𝑦 2 − 7𝑦 2 ) − 2𝑦 + 7 − 4 = (−3 + 10)𝑦 4 + (6 − 7)𝑦 2 − 2𝑦 + 3 = 7𝑦 4 − 𝑦 2 − 2𝑦 + 3 Hence, (−3𝑦 4 + 6𝑦 2 + 7) + (10𝑦 4 − 7𝑦 2 − 2𝑦 − 4) = 7𝑦 4 − 𝑦 2 − 2𝑦 + 3 Question #3 Find the LCM of the given polynomial. 4𝑥 3 − 4𝑥 2 + 𝑥, 2𝑥 3 − 𝑥 2 , 𝑥 3 Answer: Step 1 Given 4𝑥 3 − 4𝑥 2 + 𝑥, 2𝑥 3 − 𝑥 2 , 𝑥 3 Step 2 Solution Factoring them 4𝑥 3 − 4𝑥 2 + 𝑥 = 𝑥(2𝑥 − 1)2 2𝑥 3 − 𝑥 2 = 𝑥 2 (2𝑥 − 1) 𝑥3 LCM is the product of all the factors which are common, and which are not common So LCM is 𝑥 3 (2𝑥 − 1)2 Answer: 𝑥 3 (2𝑥 − 1)2 Question #4 Tell whether the expression is a polynomial. If it is, give its degree. If it is not, state why not. 3𝑥 2 − 5 Answer: Step 1 Given, 𝑓(𝑥) = 3𝑥 2 − 5 Step 2 given function 𝑓(𝑥) = 3𝑥 2 − 5 is a polynomial and degree(i.e. highest power) of f(x) is 2 Question #5 If 1 is a zero of the polynomial 𝑝(𝑥) = 𝑎𝑥 2 − 3(𝑎 − 1)𝑥 − 1, then Find the value of a Answer: Step 1 Given 𝑆𝑝(𝑥) = 𝑎𝑥 2 − 3(𝑎 − 1)𝑥 − 1 Step 2 Finding the value of a Let x be the zero Then, x=1 𝑝(1) = 𝑎(1)2 − 3(𝑎 − 1)1 − 1 =0 a-3a+3-1=0 -2a+2=0 -2a=-2 2a=2 2 𝑎= 2 a=1 ∴𝑎=1 Question #6 Find the LCM of the given polynomial. 3𝑥 2 − 27,2𝑥 2 − 𝑥 − 15 Answer: Step 1 The lowest common multiple of 2 polynomials can be obtained factorizing both the polynomials. The factors that are common are taken only once in the product term of the lowest common multiple. The factors that are not common are then multiplied with the common factors to obtain the lowest common multiple. The lowest common multiple of polynomials is also a polynomial. Step 2 The polynomial 3𝑥 2 − 27 is factorized as follows: 3𝑥 2 − 27 = 3(𝑥 2 − 9) =3(x+3)(x-3) The polynomial 2𝑥 2 − 𝑥 − 15 is factorized as follows:2𝑥 2 − 𝑥 − 15 = 2𝑥 2 − 6𝑥 + 5𝑥 − 15 =2x(x-3)+5(x-3) =(2x+5)(x-3) Step 3 The common factor between the 2 polynomials is (x-3). The other factors are obtained as 3, (x+3) and (2x+5). Thus the lowest common multiple is the product of these factors and obtained as follows:𝐿𝐶𝑀 = 3(𝑥 − 3)(𝑥 + 3)(2𝑥 + 5) = 3(𝑥 2 − 9)(2𝑥 + 5) = (3𝑥 2 − 27)(2𝑥 + 5) = 6𝑥 3 + 15𝑥 2 − 54𝑥 − 135 Thus the lowest common multiple of 3𝑥 2 − 27 and 2𝑥 2 − 𝑥 − 15 is obtained to be 6𝑥 3 + 15𝑥 2 − 54𝑥 − 135. Question #7 Factor each polynomial completely. If the polynomial cannot be factored, say it is '. 3 − 27𝑥 2 Answer: Step 1 The given polynomial is 3 − 27𝑥 2 Step 2 3 − 27𝑥 2 = 3(1 − 9𝑥 2 ) = 3(12 − (3𝑥)2 ) = 3(1 + 3𝑥)(1 − 3𝑥) Question #8 Factor each polynomial completely. If the polynomial cannot be factored, say it is '. 6𝑥 2 + 8𝑥 + 2 Answer: Given, 6𝑥 2 + 8𝑥 + 2 On simplification, we get 6𝑥 2 + 8𝑥 + 2 = 6𝑥 2 + 6𝑥 + 2𝑥 + 2 =6x(x+1)+2(x+1) =(6x+2)(x+1) =2(3x+1)(x+1) Question #9 Factor completely each polynomial, and indicate any that are not factorable using integers. 6𝑥 4 − 5𝑥 2 − 21 Answer: Step 1 Given: 6𝑥 4 − 5𝑥 2 − 21 To factor the given trinomial? Using the fact that by letting 𝑥 2 = 𝑦 (substitution method) Step 2 6𝑥 4 − 5𝑥 2 − 21 Put 𝑥 2 = 𝑦 ⇒ 6𝑦 2 − 5𝑦 − 21 = (3𝑦 − 7)(2𝑦 + 3) = (3𝑥 2 − 7)(2𝑥 2 + 3) (by replacing y as 𝑥 2 ) Therefore, 6𝑥 4 − 5𝑥 2 − 21 = (3𝑥 2 − 7)(2𝑥 2 + 3) but still it is not factored in terms of linear expression ie., the expression (2𝑥 2 + 3) can be factored into 3 3 2 2 two pairs of complex number (𝑥 + 𝑖√ ) (𝑥 − 𝐼√ ) which is absqrt in real number system here. Question #10 Factor each polynomial completely. If the polynomial cannot be factored, say it is '. 𝑥 6 − 2𝑥 3 + 1 Answer: Step 1 Factorizing a polynomial is breaking a polynomial into small polynomials. If we multiply again those smaller polynomial with each other we will get the same polynomial. There are many ways of factorizing, one such is taking GCF (greatest common factor) of each term of the polynomial and taking GCF common and represent it in terms of GCF and another factor. Another important property is distributive property which is given by a(b+c)=ab+ac. It is taking common value from function and representing it in terms of common factor and another function. Step 2 For the equation 𝑥 6 − 2𝑥 3 + 1, let's take 𝑥 3 = 𝑡 and put it in given equation to get equation in terms of variable t 𝑥 6 − 2𝑥 3 + 1 = 𝑡 2 − 2𝑡 + 1 … (1) From equation(1), factor the function of t by substituting -2t as (-t-t) and taking common values out. 𝑡 2 − 2𝑡 + 1 = 𝑡 2 − 𝑡 − 𝑡 + 1 = 𝑡(𝑡 − 1) − 1(𝑡 − 1) =(t-1)(t-1) (using distributive property) = (𝑡 − 1)2 … (2) Step 3 Resubstitute value of 𝑡 = 𝑥 3 in equation (2) to get factor form equation 𝑥 6 − 2𝑥 3 + 1 𝑡 2 − 2𝑡 + 1 = (𝑥 3 − 1)2 𝑥 6 − 2𝑥 3 + 1 = (𝑥 3 − 1)2 (from equation(1)) Therefore factored form of 𝑥 6 − 2𝑥 3 + 1 is (𝑥 3 − 1)2