Questions and Answers #20 Polynomial Factorization

Questions and Answers Sheet 20 Polynomial Factorization Question #1 Factor each polynomial by grouping. 𝑝2 𝑞 2 − 10 − 2𝑞 2 + 5𝑝2 Answer: Step 1 The given polynomial is 𝑝2 𝑞 2 − 10 − 2𝑞 2 + 5𝑝2 To factor by grouping, we can group the terms as follows, (𝑝2 𝑞 2 − 2𝑞 2 ) + (5𝑝2 − 10) Step 2 Now we will factorize this expression, (𝑝2 𝑞 2 − 2𝑞 2 ) + (5𝑝2 − 10) In the first bracket, we have 𝑞 2 as common factor and in the second bracket, we have 5 as a common factor. Factoring out the common factor we get, 𝑝2 (𝑞 2 − 2) + 5(𝑞 2 − 2) Next, we have (𝑞 2 − 2) as a common factor in the whole expression. Factoring it out we get, (𝑞 2 − 2)(𝑝2 + 5) This will be the factorization of the given expression using the grouping method. Question #2 Solve the given quadratic equation by factorization method. 𝑥+3 𝑥−2 − 1−𝑥 𝑥 = 17 4 Answer: Step 1 Given 𝑥+3 1−𝑥 17 − = 𝑥−2 𝑥 4 Step 2 Finding the value of x 𝑥(𝑥+3)−(1−𝑥)(𝑥−2) = 𝑥(𝑥−2) 𝑥 2 +3𝑥−(𝑥−2−𝑥 2 +2𝑥) 𝑥 2 −2𝑥 𝑥 2 +3𝑥−𝑥+2+𝑥 2 −2𝑥 2𝑥 2 +2 𝑥 2 −2𝑥 2 𝑥 2 −2𝑥 17 = 4 17 4 = = 17 4 17 4 8𝑥 + 8 = 17𝑥 2 − 34𝑥 17𝑥 2 − 34𝑥 − 8𝑥 2 − 8 = 0 9𝑥 2 − 34𝑥 − 8 = 0 9𝑥 2 − 36𝑥 + 2𝑥 − 8 = 0 9𝑥(𝑥 − 4) + 2(𝑥 − 4) = 0 (9𝑥 + 2)(𝑥 − 4) = 0 9𝑥 + 2 = 0; 𝑥 − 4 = 0 9𝑥 = −2; 𝑥 = 4 −2 9 −2 ∴ 𝑥 = 4 or 𝑥 = 𝑥= 9 Question #3 What is the correct factorization of the expression 2𝑥 2 + 7𝑥 − 15 Answer: Step 1 Explanation of the answer is as follows Step 2 Solution: given that 2𝑥 2 + 7𝑥 − 15 = 2𝑥 2 + 10𝑥 − 3𝑥 − 15 = 2𝑥(𝑥 + 5) − 3(𝑥 + 5) = (𝑥 + 5)(2𝑥 − 3) Hence factor of 2𝑥 2 + 7𝑥 − 15 is (𝑥 + 5)(2𝑥 − 3) Question #4 Solve the Quadratic equation by factorization 𝑥 2 − 5𝑥 − 24 = 0 Answer: Step 1 given that, 𝑥 2 − 5𝑥 − 24 = 0 we have to solve the quadratic equation by factorization Step 2 here, we using the splitting method to factorize the quadratic equation in order to factorize, 𝑥 2 − 5𝑥 − 24 = 0, we find two numbers p and q such that p-q=5 and 𝑝 × 𝑞 = 24 . clearly, 8-3=5, 8 × 3 = 24 now we split the middle term 𝑥 2 − 5𝑥 − 24 = 0 ⇒ 𝑥 2 − (8 − 3)𝑥 − 24 = 0 ⇒ 𝑥 2 − 8𝑥 + 3𝑥 − 24 = 0 ⇒ 𝑥(𝑥 − 8) + 3(𝑥 − 8) = 0 ⇒ (𝑥 − 8)(𝑥 + 3) = 0 ⇒ 𝑥 − 8 = 0, 𝑥 + 3 = 0 ⇒ 𝑥 = 8, 𝑥 = −3 Question #5 Find the roots of the given quadratic equation by factorization method 𝑥 2 − 3𝑥 − 10 = 0 Answer: Step 1 Given 𝑥 2 − 3𝑥 − 10 = 0 Step 2 Finding the roots 𝑥 2 − 5𝑥 + 2𝑥 − 10 𝑥(𝑥 − 5) + 2(𝑥 − 5) = 0 (𝑥 − 5)(𝑥 + 2) = 0 𝑥 − 5 = 0; 𝑥 + 2 = 0 𝑥 = 5; 𝑥 = −2 𝑥 = 5, −2 ∴ Roots are 5,-2 Question #6 Solve the following differential equation by factorization method. 𝑥 2 − 2𝑎𝑥 + 𝑎2 − 𝑏 2 = 0 Answer: Step 1 The given quadratic equation is, 𝑥 2 − 2𝑎𝑥 + 𝑎2 − 𝑏 2 = 0 Step 2 2a is written as, 2a=(a+b)+(a-b) and 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) 𝑥 2 − ((𝑎 + 𝑏) + (𝑎 − 𝑏))𝑥 + (𝑎 + 𝑏)(𝑎 − 𝑏) = 0 𝑥 2 − (𝑎 + 𝑏)𝑥 − (𝑎 − 𝑏)𝑥 + (𝑎 + 𝑏)(𝑎 − 𝑏) = 0 x(x-(a+b))-(a-b)(x-(a+b)) (x-(a+b))(x-(a-b)) x-(a+b)=0, x-(a-b)=0 x=(a+b), x=(a-b) x=a+b, x=a-b This is the solution of the quadratic equation is, 𝑥 2 − 2𝑎𝑥 + 𝑎2 − 𝑏 2 = 0 Question #7 Complete the factorization. 2𝑡 2 (6𝑡 − 1) + 3(6𝑡 − 1) Answer: Step 1 We have 2𝑡 2 (6𝑡 − 1) + 3(6𝑡 − 1) Step 2 Taking (6t-1) as a common in above term, we get (6𝑡 − 1)(2𝑡 2 + 3) Step 3 Answer: (6𝑡 − 1)(2𝑡 2 + 3) Question #8 Solve the following quadratic equation by factorization method 4𝑥 2 − 4𝑎𝑥 + (𝑎2 − 𝑏 2 ) = 0 Answer: Step 1 Given 4𝑥 2 − 4𝑎𝑥 + (𝑎2 − 𝑏 2 ) = 0 Step 2 solving Here, constant term = (𝑎2 − 𝑏 2 ) = (𝑎 − 𝑏)(𝑎 + 𝑏) and, Coefficient of middle term =-4a Also, Coefficient of the middle term −4𝑎 = −[2(𝑎 + 𝑏) + 2(𝑎 − 𝑏)] ∴ 4𝑥 2 − 4𝑎𝑥 + (𝑎2 − 𝑏 2 ) = 0 4𝑥 2 − {2(𝑎 + 𝑏) + 2(𝑎 − 𝑏)}𝑥 + (𝑎 + 𝑏)(𝑎 − 𝑏) = 0 4𝑎2 − 2(𝑎 + 𝑏)𝑥 − 2(𝑎 − 𝑏)𝑥 + (𝑎 + 𝑏)(𝑎 − 𝑏) = 0 {4𝑥 2 − 2(𝑎 + 𝑏)𝑥} − {2(𝑎 − 𝑏)𝑥 − (𝑎 + 𝑏)(𝑎 − 𝑏)} = 0 2𝑥(2𝑥 − (𝑎 + 𝑏) − (𝑎 − 𝑏)){2𝑥 − (𝑎 + 𝑏)} = 0 {2𝑥 − (𝑎 + 𝑏)}{2𝑥 − (𝑎 − 𝑏)} = 0 {2𝑥 − (𝑎 + 𝑏)} = 0 𝑜𝑟, {2𝑥 − (𝑎 − 𝑏)} = 0 2x=a+b or, 2x=a-b 𝑥= 𝑎+𝑏 2 or, 𝑥 = 𝑎−𝑏 2 . Question #9 Solve the following quadratic equation by factorization method: 𝑥 2 + 3𝑥 − (𝑎2 + 𝑎 − 2) = 0 Answer: Step 1 To solve the quadratic equation using factorization method. Step 2 𝑥 2 + 3𝑥 − (𝑎2 + 𝑎 − 2) = 0 𝑥 2 + 3𝑥 − (𝑎2 + 2𝑎 − 𝑎 − 2) = 0 𝑥 2 + 3𝑥 − (𝑎(𝑎 + 2) − 1(𝑎 + 2)) = 0 𝑥 2 + 3𝑥 − ((𝑎 + 2)(𝑎 − 1)) = 0 𝑥 2 + ((𝑎 + 2) − (𝑎 − 1))𝑥 − ((𝑎 + 2)(𝑎 − 1)) = 0 𝑥 2 + (𝑎 + 2)𝑥 − (𝑎 − 1)𝑥 − (𝑎 + 2)(𝑎 − 1) = 0 𝑥(𝑥 + (𝑎 + 2)) − (𝑎 − 1)(𝑥 + (𝑎 + 2)) = 0 (𝑥 + (𝑎 + 2))(𝑥 − (𝑎 − 1)) = 0 x+(a+2)=0 or x-(a-1)=0 x=-a-2, x=a-1 Question #10 Which of the following is the correct factorization of 𝑥 2 + 8𝑥 + 15? Answer: Step 1 Find Correct Factorization. Step 2 𝑥 2 + 8𝑥 + 15 𝑥 2 + 5𝑥 + 3𝑥 + 15 𝑥(𝑥 + 5) + 3(𝑥 + 5) (𝑥 + 3)(𝑥 + 5) (Answer)