Questions and Answers #75 Polynomial Factorization

Questions and Answers Sheet 75 Polynomial Factorization Question #1 How do you combine (𝑥 2 − 𝑏𝑥 − 7) − (3𝑥 2 + 2𝑥 − 4)? Answer: Group by like powers of x and perform standard arithmetic to get: −2𝑥 2 − (𝑏 + 2)𝑥 − 3 Explanation: (𝑥 2 − 𝑏𝑥 − 7) − (3𝑥 2 + 2𝑥 − 4) Regroup as: (𝑥 2 − 3𝑥 2 ) + (−𝑏𝑥 − 2𝑥) + (−7 − (−4)) = −2𝑥 2 − (𝑏 + 2)𝑥 − 3 Question #2 What is the standard form of 𝑦 = (2𝑥 − 3)(2𝑥 + 4) + (4𝑥 − 3)2 ? Answer: Explanation: 𝑦 = (2𝑥 − 3)(2𝑥 + 4) + (4𝑥 − 3)2 by distributive property and using special product 𝑦 = 4𝑥 2 + 8𝑥 − 6𝑥 − 12 + 16𝑥 2 − 24𝑥 + 9 𝑦 = 20𝑥 2 − 22𝑥 − 3 Answer: 𝑦 = 20𝑥 2 − 22𝑥 − 3 Question #3 How do you subtract (𝑥 3 − 𝑥 − 4) − (𝑥 2 + 𝑥 − 4)? Answer: Explanation: First, remove all of the terms from parenthesis. Be careful to handle the signs of each individual term correctly: 𝑥3 − 𝑥 − 4 − 𝑥2 − 𝑥 + 4 Next, group like terms: 𝑥3 − 𝑥2 − 𝑥 − 𝑥 − 4 + 4 Now, combine like terms: 𝑥 3 − 𝑥 2 − 1𝑥 − 1𝑥 − 4 + 4 𝑥 3 − 𝑥 2 + (−1 − 1)𝑥 + (−4 + 4) 𝑥 3 − 𝑥 2 + (−2)𝑥 + 0 𝑥 3 − 𝑥 2 − 2𝑥 This can be factored as: 𝑥(𝑥 − 2)(𝑥 + 1) Question #4 What is the standard form of 𝑦 = (2𝑥 − 9)(𝑥 − 5) − (2𝑥 + 7)2 ? Answer: Explanation: The general standard form for a quadratic is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 with constants a,b,c Given 𝑦 = (2𝑥 − 9)(𝑥 − 5) − (2𝑥 + 7)2 Expanding the terms: 𝑦 = (2𝑥 2 − 19𝑥 + 45) − (4𝑥 2 + 28𝑥 + 49) Combine like terms: 𝑦 = −2𝑥 2 − 47𝑥 − 4 Question #5 What is the standard form of 𝑦 = (𝑥 − 7)(3𝑥 − 5) − (𝑥 + 7)2 ? Answer: Explanation: Consider the first part: (𝑥 − 7)(3𝑥 − 5) → 3𝑥 2 − 5𝑥 − 21𝑥 + 35 = 3𝑥 2 − 26𝑥 + 35 Consider the second part: (𝑥 − 7)2 → 𝑥 2 − 14𝑥 + 49 Put them together and we end up with: 𝑦 = (3𝑥 2 − 26𝑥 + 35) − (𝑥 2 − 14𝑥 + 49) The minus sign outside the brackets reverses all the signs within. 𝑦 = 3𝑥 2 − 26𝑥 + 35 − 𝑥 2 + 14𝑥 − 49 𝑦 = 2𝑥 2 − 12𝑥 − 14 So the standard form is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Question #6 What is the standard form of 𝑦 = (𝑥 + 7)(2𝑥 + 15) − (𝑥 − 7)2 ? Answer: Explanation: standard form is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 first multiply/distribute to expand everything: 𝑦 = (𝑥 + 7)(2𝑥 + 15) − (𝑥 − 7)2 𝑦 = 𝑥(2𝑥 + 15) + 7(2𝑥 + 15) − (𝑥 − 7)(𝑥 − 7) 𝑦 = 2𝑥 2 + 15𝑥 + 14𝑥 + 105 − (𝑥(𝑥 − 7) − 7(𝑥 − 7)) 𝑦 = 2𝑥 2 + 29𝑥 + 105 − (𝑥 2 − 7𝑥 − 7𝑥 + 49) combine like terms as you go 𝑦 = 2𝑥 2 + 29𝑥 + 105 − 𝑥 2 + 14𝑥 − 49 𝑦 = 𝑥 2 + 43𝑥 + 56 Question #7 What is the standard form of 𝑦 = (𝑥 + 3)(𝑥 + 1) + (3𝑥 − 7)2 ? Answer: Explanation: As this a quadratic equation, standard form of this is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Hence simplifying 𝑦 = (𝑥 + 3)(𝑥 + 1) + (3𝑥 − 7)2 = (𝑥 2 + 3𝑥 + 𝑥 + 3) + ((3𝑥)2 + 2 × 3 × 𝑥(−7) + 72 ) = (𝑥 2 + 3𝑥 + 𝑥 + 3) + (9𝑥 2 − 42𝑥 + 49) = 10𝑥 2 − 38𝑥 + 52 Question #8 What is the standard form of 𝑦 = (𝑥 + 3)(−𝑥 − 1) + (3𝑥 − 7)2 ? Answer: Explanation: It is apparent that the highest degree of x in the function (𝑥 + 3)(−𝑥 − 1) + (3𝑥 − 7)2 is two. Expanding the function (𝑥 + 3)(−𝑥 − 1) + (3𝑥 − 7)(3𝑥 − 7) (𝑥 + 3)(−𝑥) − 1(𝑥 + 3) + 3𝑥(3𝑥 − 7) − 7(3𝑥 − 7)𝑜𝑟 −𝑥 2 − 3𝑥 − 𝑥 − 3 + 9𝑥 2 − 21𝑥 − 21𝑥 + 49𝑜𝑟 8𝑥 2 − 46𝑥 + 46 As it is an function in degree 2 of form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Question #9 What is the standard form of 𝑦 = (𝑥 + 4)(𝑥 + 11) − (5𝑥 − 7)2 ? Answer: Explanation: 𝑦 = (𝑥 + 4)(𝑥 + 11) − (5𝑥 − 7)2 𝑦 = 𝑥 2 + 4𝑥 + 11𝑥 + 44 − 25𝑥 2 + 70𝑥 − 49 𝑦 = −24𝑥 2 + 85𝑥 − 5 Standard form is 𝑦 = −24𝑥 2 + 85𝑥 − 5 Question #10 What is the standard form of a polynomial (9𝑎2 − 4 − 5𝑎) − (12𝑎 − 6𝑎2 + 3)? Answer: Explanation: First, remove all of the terms from parenthesis. Be careful to handle the signs of each individual term correctly: 9𝑎2 − 4 − 5𝑎 − 12𝑎 + 6𝑎2 − 3 Next, group like terms in descending order of the power of their exponents: 9𝑎2 + 6𝑎2 − 5𝑎 − 12𝑎 − 4 − 3 Now, combine like terms: (9 + 6)𝑎2 + (−5 − 12)𝑎 + (−4 − 3) 15𝑎2 + (−17)𝑎 + (−7) 15𝑎2 − 17𝑎 − 7