Questions and Answers #28 Polynomial Factorization

Questions and Answers Sheet 28 Polynomial Factorization Question #1 Factor each polynomial. 𝑥 2 − 4𝑥 − 21 Answer: Step 1 Consider the polynomial: 𝑥 2 − 4𝑥 − 21 Assume that the above equation is in the form 𝑥 2 + 𝑏𝑥 + 𝑐 where b is sum of factors and c is product of factors Find a pair of numbers whose product is and whose sum is. In this case, whose product is and whose sum is -7,3 Step 2 Write the factored form using these integers: (𝑥 − 7)(𝑥 + 3) Question #2 Factor the polynomial. 21𝑥 2 + 41𝑥 + 10 = 0 Answer: Step 1 Given: 21𝑥 2 + 41𝑥 + 10 = 0 Step 2 21𝑥 2 + 41𝑥 + 10 = 0 compare given a polynomial with 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 𝑎 = 21, 𝑏 = 41, 𝑐 = 10 Step 3 Formula: 𝑥= −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 Step 4 𝑥= = = = −41±√412 −4(21)(10) 2(21) −41±√1681−840 2(21) −41±√841 2(21) −41±29 𝑥= 42 −41+29 42 and 𝑥 = −41−29 Step 5 𝑥= 𝑥= 12 42 2 6 and 𝑥 = and 𝑥 = −70 6 42 10 42 𝑥= 1 3 1 and 𝑥 = 3 5 3 (𝑥 − ) = 0 and (𝑥 − ) = 0 3 5 Question #3 Factor the polynomial. 𝑥 2 + 3𝑥 − 4 Answer: Step 1 we have to factor the polynomial: 𝑥 2 + 3𝑥 − 4 Step 2 here, 𝑥 2 + 3𝑥 − 4 𝑥 2 + 3𝑥 − 4 = 𝑥 2 + (4 − 1)𝑥 − 4 = 𝑥 2 + 4𝑥 − 𝑥 − 4 = 𝑥(𝑥 + 4) − 1(𝑥 + 4) = (𝑥 + 4)(𝑥 − 1) ⇒ 𝑥 2 + 3𝑥 − 4 = (𝑥 + 4)(𝑥 − 1) this is the required answer. Question #4 Factor the polynomial. 6𝑥 2 + 7𝑥 − 20 Answer: Step 1 We have to factorize the polynomial: 6𝑥 2 + 7𝑥 − 20 Factorizing the polynomial by middle term splitting method Here, 6 × (−20) = −120 = 15 × (−8) For middle term, 15 − 8 = 7 Step 2 So using above condition factorizing the given polynomial, 6𝑥 2 + 7𝑥 − 20 = 6𝑥 2 + 15𝑥 − 8𝑥 − 20 = 3𝑥(2𝑥 + 5) − 4(2𝑥 + 5) = (2𝑥 + 5)(3𝑥 − 4) Hence, factorized form of the polynomial is (2𝑥 + 5)(3𝑥 − 4). Question #5 Factor the polynomial. 9𝑦 4 − 121𝑥 2 Answer: Step 1 The polynomial is 9𝑦 4 − 121𝑥 2 . Rewrite the expression using the exponent rule as follows. 9𝑦 4 − 121𝑥 2 = (3𝑦 2 )2 − (11𝑥)2 Step 2 Apply the difference of two squares formula 𝑥 2 − 𝑦 2 = (𝑥 + 𝑦)(𝑥 − 𝑦) and factor the polynomial where 𝑥 = 3𝑦 2 and 𝑦 = 11𝑥. 9𝑦 4 − 121𝑥 2 = (3𝑦 2 )2 − (11𝑥)2 = (3𝑦 2 + 11𝑥)(3𝑦 2 − 11𝑥) Therefore, 9𝑦 4 − 121𝑥 2 = (3𝑦 2 + 11𝑥)(3𝑦 2 − 11𝑥). Question #6 Factor each polynomial. 25𝑠 4 − 9𝑡 2 Answer: Step 1 Given, 25𝑠 4 − 9𝑡 2 Step 2 On simplification, we get 25𝑠 4 − 9𝑡 2 = (5𝑠 2 )2 − (3𝑡)2 = (5𝑠 2 − 3𝑡)(5𝑠 2 + 3𝑡) [∵ 𝑎2 − 𝑏 2 = (𝑎 − 𝑏)(𝑎 + 𝑏)] Question #7 Factor each polynomial. 6𝑧 4 − 13𝑧 2 − 5 Answer: Step 1 Given polynomial is: 6𝑧 4 − 13𝑧 2 − 5 Step 2 6𝑧 4 − 13𝑧 2 − 5 = 6𝑧 4 − 15𝑧 2 + 2𝑧 2 − 5 = 3𝑧 2 (2𝑧 2 − 5) + 1(2𝑧 2 − 5) = (3𝑧 2 + 1)(2𝑧 2 − 5) are the required factors. Question #8 Add polynomials: −3𝑥 2 + 2𝑥 + 6 and −𝑥 2 − 𝑥 − 1. Answer: Step 1 Given: −3𝑥 2 + 2𝑥 + 6 and −𝑥 2 − 𝑥 − 1 . Calculation: 𝑓(𝑥) = −3𝑥 2 + 2𝑥 + 6 𝑔(𝑥) = −𝑥 2 − 𝑥 − 1 Step 2 𝑓(𝑥) + 𝑔(𝑥) = −3𝑥 2 + 2𝑥 + 6 + (−𝑥 2 − 𝑥 − 1) = −3𝑥 2 + 2𝑥 + 6 − 𝑥 2 − 𝑥 − 1 = −3𝑥 2 − 𝑥 2 + 2𝑥 − 𝑥 + 6 − 1 𝑓(𝑥) + 𝑔(𝑥) = −4𝑥 2 + 𝑥 + 5 Question #9 Factor the polynomial: 3𝑥 2 + 18𝑥 + 27 Answer: Step 1 Given, 3𝑥 2 + 18𝑥 + 27 Step 2 On simplification, we get 3𝑥 2 + 18𝑥 + 27 = 3(𝑥 2 + 6𝑥 + 9) = 3(𝑥 2 + 3𝑥 + 3𝑥 + 9) = 3[𝑥(𝑥 + 3) + 3(𝑥 + 3)] = 3[(𝑥 + 3)(𝑥 + 3)] = 3(𝑥 + 3)2 Question #10 Factor the polynomial 36𝑎2 − 84𝑎𝑡 + 49𝑡 2 Answer: Step 1 Consider the given expression. 36𝑎2 − 84𝑎𝑡 + 49𝑡 2 Step 2 On simplifying the above expression, then ⇒ (6𝑎)2 − 2 × 6𝑎 × 7𝑡 + (7𝑡)2 ⇒ (6𝑎)2 + (7𝑡)2 − 2 × 6𝑎 × 7𝑡 Step 3 Property used: (𝑥 − 𝑦)2 = 𝑥 2 + 𝑦 2 − 2𝑥𝑦 Step 4 Therefore, the factor of the given expression will be ⇒ (6𝑎 − 7𝑡)2 ⇒ (6𝑎 − 7𝑡)(6𝑎 − 7𝑡) … . [∵ (𝑥 − 𝑦)2 = (𝑥 − 𝑦)(𝑥 − 𝑦)] Step 5 Answer: The factor of the given expression is equal to (6𝑎 − 7𝑡)(6𝑎 − 7𝑡).