Questions and Answers #53 Polynomial Factorization

Questions and Answers Sheet 53 Polynomial Factorization Question #1 Find the LCM of the given polynomial. 𝑥 3 − 𝑥, 𝑥 3 − 2𝑥 2 + 𝑥, 𝑥 3 − 1 Answer: Step 1 Given: 𝑥 3 − 𝑥, 𝑥 3 − 2𝑥 2 , 𝑥3 − 1 Step 2 𝑥 3 − 𝑥 = 𝑥(𝑥 2 − 1) = 𝑥(𝑥 2 − 12 ) = 𝑥(𝑥 + 1)(𝑥 − 1) 𝑥 3 − 2𝑥 2 = 𝑥 2 (𝑥 − 2) 𝑥 3 − 1 = 𝑥 3 − 13 = (𝑥 − 1)(𝑥 2 + 𝑥 + 1) Hence, 𝐿𝐶𝑀 = 𝑥 2 (𝑥 + 1)(𝑥 − 1)(𝑥 − 2)(𝑥 2 + 𝑥 + 1) Question #2 If a polynomial p(x) is divided with a polynomial (x + 3) using long division and the quotient is 2𝑥 2 𝑥 + 1. Find p(x). Answer: Step 1 Given that: If a polynomial p(x) is divided with a polynomial (x + 3) using long division and the quotient is 2𝑥 2 + 𝑥 + 1. Find p(x). Step 2 Formula Used: Dividend = Divisior × Quotient + Remainder or 𝑝(𝑥) = 𝑔(𝑥) × 𝑞(𝑥) + 𝑟(𝑥) Step 3 Calculation We have, g(x)=x+3 𝑞(𝑥) = 2𝑥 2 + 𝑥 + 1 r(x)=0 So by the formula used we have, 𝑝(𝑥) = (𝑥 + 3)(2𝑥 2 + 𝑥 + 1) + 0 = 2𝑥 3 + 𝑥 2 + 𝑥 + 6𝑥 2 + 3𝑥 + 3 = 2𝑥 3 + 7𝑥 2 + 4𝑥 + 3 Question #3 Factor completely each polynomial, and indicate any that are not factorable using integers. 2𝑛3 + 14𝑛2 − 20𝑛 Answer: Step 1 Given 2𝑛3 + 14𝑛2 − 20𝑛 We have to factorize the given polynomial. Step 2 Calculation: 2𝑛3 + 14𝑛2 − 20𝑛 = 2𝑛(𝑛2 + 7𝑛 − 10) Question #4 Multiply the polynomials using the special product formulas. (3𝑥 − 2)3 Answer: Step 1 Multiply the polynomials using the special product formulas. (3𝑥 − 2)2 Step 2 Now, (3𝑥 − 2)3 = (3𝑥)3 − 3 ∗ (3𝑥)2 + 3 ∗ 3𝑥 ∗ 22 − 23 (∵ (𝑎 − 𝑏)3 = 𝑎3 − 3𝑎2 𝑏 + 3𝑎𝑏 2 − 𝑏 3 ) = 27𝑥 3 − 3(9𝑥 2 ) ∗ 2 + 9𝑥 ∗ 4 − 8 = 27𝑥 3 − 54𝑥 2 + 36𝑥 − 8 ∴ (3𝑥 − 2)3 = 27𝑥 3 − 54𝑥 2 + 36𝑥 − 8 Question #5 Factor out the greatest common factor from each polynomial. 4𝑘 2 𝑚3 + 8𝑘 4 𝑚3 − 12𝑘 2 𝑚4 Answer: Step 1 We need to find out the greatest common factor of the above equation. Step 2 4𝑘 2 𝑚3 + 8𝑘 4 𝑚3 − 12𝑘 2 𝑚4 we need to factor out greatest common factor. 4𝑘 2 𝑚3 [1 + 2𝑘 − 3𝑚] ∴ greatest common factor is 4𝑘 2 𝑚3 Question #6 Multiply the polynomials using the special product formulas. (𝑥 + 5)2 Answer: Product of (𝑥 + 5)2 (𝑥 + 5)2 = (𝑥 + 5)(𝑥 + 5) using identities (𝑥 + 𝑎)(𝑥 + 𝑏) = 𝑥 2 + (𝑎 + 𝑏)𝑥 + 𝑎𝑏 (𝑥 + 5)(𝑥 + 5) = 𝑥 2 + (5 + 5)𝑥 + 5 × 5 = 𝑥 2 + 10𝑥 + 25 OR (𝑥 + 5)2 using identity (𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2 (𝑥 + 5)2 = 𝑥 2 + 2𝑥 × 5 + (5)2 = 𝑥 2 + 10𝑥 + 25 ∴ Polynomial = 𝑥 2 + 10𝑥 + 25 Question #7 For the following exercises, find the degree and leading coefficient for the given polynomial. 𝑥(4 − 𝑥 2 )(2𝑥 + 1) Answer: Step 1 Given expression: 𝑥(4 − 𝑥 2 )(2𝑥 + 1) Step 2 Solution: (4𝑥 − 𝑥 3 )(2𝑥 + 1) 2𝑥(4𝑥 − 𝑥 3 ) + 1(4𝑥 − 𝑥 3 ) 8𝑥 2 − 2𝑥 4 + 4𝑥 − 𝑥 3 −2𝑥 4 − 𝑥 3 + 8𝑥 2 + 4𝑥 Thus degree of polynomials is = 4 Leading coefficient = -2 Question #8 Multiply the polynomials using the FOIL method. (𝑥 − 5)(𝑥 − 1) Answer: Step 1 We have to find the product of the following polynomial: (𝑥 − 5)(𝑥 − 1) Step 2 Multiplying throughout we get, (𝑥 − 5)(𝑥 − 1) = 𝑥 2 − 5𝑥 − 𝑥 + 5 = 𝑥 2 − 6𝑥 + 5 Hence the answer. Question #9 Find the LCM of the given polynomial. 𝑥 2 + 4𝑥 + 4, 𝑥 3 + 2𝑥 2 , (𝑥 + 2)3 Answer: Step 1 The given polynomials are, 𝑥 2 + 4𝑥 + 4, 𝑥 3 + 2𝑥 2 , (𝑥 + 2)3 We need to find the LCM of the given polynomials. Step 2 On factorizing the first polynomial, we get 𝑥 2 + 4𝑥 + 4 = 𝑥 2 + 2𝑥 ∗ 2 + 22 = (𝑥 + 2)2 On factorizing the second polynomial, we get 𝑥 3 + 2𝑥 2 = 𝑥 2 (𝑥 + 2) And the third polynomial is (𝑥 + 2)3 The lowest common multiple of the three polynomials is the polynomial which is multiple of the given three polynomials, Therefore, the LCM of the polynomials is 𝑥 2 (𝑥 + 2)3 Question #10 What is the degree of the polynomial −3𝑥 3 𝑦 2 + 3𝑥 2 𝑦 4 + 3𝑥𝑦 2 Answer: Step 1 Rewrite the given polynomial in terms of decreasing powers. −3𝑥 3 𝑦 2 + 3𝑥 2 𝑦 4 + 3𝑥𝑦 2 = 3𝑥 2 𝑦 4 − 3𝑥 3 𝑦 2 + 3𝑥𝑦 2 Step 2 Now degree of the polynomial can be determined to be the largest sum of powers of variables in the given expression which is determined to be 6 in the given polynomial as. 3𝑥 2 𝑦 4 degree = 2+4=6