Questions and Answers #26 Polynomial Factorization

Questions and Answers Sheet 26 Polynomial Factorization Question #1 To add polynomials, we add _______ terms. So (3𝑥 2 + 2𝑥 + 4) + (8𝑥 2 − 𝑥 + 1) = ______ Answer: Step 1 We have to fill in the bla: To add polynomials, we add ________ terms. So (3𝑥 2 + 2𝑥 + 4) + (8𝑥 2 − 𝑥 + 1) = _______. We know that to add polynomial we add similar terms. Example: (𝑥 + 2) + (𝑥 2 + 3𝑥 + 4) = 𝑥 + 2 + 𝑥 2 + 3𝑥 + 4 = 𝑥 2 + 3𝑥 + 𝑥 + 2 + 4 = 𝑥 2 + 4𝑥 + 6 Here we have combined the terms x with x and constants with constants. Step 2 Applying addition property of the polynomial for the given expression, we get (3𝑥 2 + 2𝑥 + 4) + (8𝑥 2 − 𝑥 + 1) = 3𝑥 2 + 2𝑥 + 4 + 8𝑥 2 − 𝑥 + 1 = 3𝑥 2 + 8𝑥 2 + 2𝑥 − 𝑥 + 4 + 1 = 11𝑥 2 + 𝑥 + 5 Hence, we can fill in the bla: To add polynomials, we add similar terms. So (3𝑥 2 + 2𝑥 + 4) + (8𝑥 2 − 𝑥 + 1) = 11𝑥 2 + 𝑥 + 5. Question #2 Which of the following is the correct factorization of 27𝑥 3 − 1 ? (3𝑥 − 1)(3𝑥 2 + 𝑥 + 1) (3𝑥 + 1)(9𝑥 2 − 3𝑥 + 1) (3𝑥 − 1)(9𝑥 2 + 3𝑥 + 1) (3𝑥 − 1)(9𝑥 2 + 3𝑥 − 1) Answer: Step 1 which of the following is the correct factorization of 27𝑥 3 − 1 ? ⇒ 27𝑥 3 − 1 ⇒ (3𝑥)3 − (1)3 Step 2 ⇒ 27𝑥 3 − 1 ⇒ (3𝑥)3 − (1)3 Use 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑏 2 + 𝑎𝑏) ⇒ (3𝑥)3 − (1)3 ⇒ (3𝑥 − 1)[(3𝑥)2 + (1)2 + 3𝑥 × 1] 27𝑥 3 − 1 ⇒ (3𝑥 − 1)[9𝑥 2 + 1 + 3𝑥] (Option C) Question #3 Factor the polynomial. 64𝑥 3 + 27 Answer: Step 1 To factor the polynomial: 64𝑥 3 + 27 Use the identity: 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 ) Step 2 Now, 64𝑥 3 + 27 = (4𝑥)3 + 33 Here, a=4x and b=3 64𝑥 3 + 27 = (4𝑥)2 + 33 = (4𝑥 + 3)((4𝑥)2 − (4𝑥)(3) + 32 ) = (4𝑥 + 3)(16𝑥 2 − 12𝑥 + 9) Question #4 Factor the polynomial. 125𝑥 3 − 8 Answer: Step 1 Given expression is (125𝑥 3 − 8) Since, both terms are perfect cubes. We will use the formula (𝑎3 − 𝑏 3 ) = (𝑎 − 𝑏)(𝑎2 + 𝑏 2 + 𝑎𝑏) Here, a=5x and b=2 We can write as (5𝑥)3 − (2)3 = (5𝑥 − 2)[(5𝑥)2 + 22 + 5𝑥 × 2] = (5 − 2𝑥)(25𝑥 2 + 4 + 10𝑥) Step 2 Hence, (125𝑥 3 − 8) = (5𝑥 − 2)(25𝑥 2 + 10𝑥 + 4) Question #5 Factor each polynomial. 9𝑎2 − 16 Answer: Step 1 Formula: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Step 2 9𝑎2 − 16 = (3𝑎)2 = (4)2 = (3𝑎 + 4)(3𝑎 − 4) Question #6 Factor each polynomial. 𝑥 2 − 8𝑥 + 16 − 𝑦 2 Answer: Step 1 We need to factor the following polynomial 𝑥 2 − 8𝑥 + 16 − 𝑦 2 Step 2 The polynomial can be factored as following 𝑥 2 − 8𝑥 + 16 − 𝑦 2 = (𝑥 2 − 8𝑥 + 16) − 𝑦 2 = (𝑥 − 4)2 − 𝑦 2 = [(𝑥 − 4) − 𝑦][(𝑥 − 4) + 𝑦] Using formula 𝑎2 − 𝑏 2 = (𝑎 − 𝑏)(𝑎 + 𝑏) = (𝑥 − 4 − 𝑦)(𝑥 − 4 + 𝑦) Hence, 𝑥 2 − 8𝑥 + 16 − 𝑦 2 = (𝑥 − 4 − 𝑦)(𝑥 − 4 + 𝑦) Question #7 Factor each polynomial. 𝑚2 + 10𝑚 + 25 − 𝑛4 Answer: Step 1 Given 𝑚2 + 10𝑚 + 25 − 𝑛4 To fins the factor of the polynomial Step 2 𝑚2 + 10𝑚 + 25 − 𝑛4 ⇒ (𝑚2 + 10𝑚 + 25) − 𝑛4 ⇒ (𝑚 + 5)2 − 𝑛4 Since both terms are perfect squares factor using the difference formula 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) ⇒ (𝑚 + 5 − 𝑛2 )(𝑚 + 5 + 𝑛2 ) Answer: (𝑚 + 5 − 𝑛2 )(𝑚 + 5 + 𝑛2 ) Question #8 Factor the polynomial. 𝑥 16 − 1 Answer: Step 1 To factor the polynomial: 𝑥 16 − 1 Solution: We have identity 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Using the above identity, we will factor the given expression. 𝑥 16 − 1 = (𝑥 8 )2 − 12 = (𝑥 8 + 1)(𝑥 8 − 1) = (𝑥 8 + 1)((𝑥 4 )2 − 12 ) = (𝑥 8 + 1)(𝑥 4 + 1)(𝑥 4 − 1) = (𝑥 8 + 1)(𝑥 4 + 1)((𝑥 2 )2 − 12 ) = (𝑥 8 + 1)(𝑥 4 + 1)(𝑥 2 + 1)(𝑥 2 − 1) = (𝑥 8 + 1)(𝑥 4 + 1)(𝑥 2 + 1)(𝑥 + 1)(𝑥 − 1) Step 2 Hence, required answer is 𝑥 16 − 1 = (𝑥 8 + 1)(𝑥 4 + 1)(𝑥 2 + 1)(𝑥 + 1)(𝑥 − 1). Question #9 Factor the polynomial 64 − 49𝑥 3 Answer: Step 1 Given 64𝑥 − 49𝑥 2 Step 2 = 𝑥(64 − 49𝑥 2 ) = 𝑥((8)2 − (7𝑥)2 ) using (𝑎2 − 𝑏 2 ) = (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑥(8 − 7𝑥)(8 + 7𝑥) finally 64𝑥 − 49𝑥 3 = 𝑥(8 − 7𝑥)(8 + 7𝑥) Question #10 In the following, add or subtract the polynomials. (8𝑚2 + 12𝑚 − 5) − (2𝑚2 − 7𝑚 − 1) Answer: Step 1 Given (8𝑚2 + 12𝑚 − 5) − (2𝑚2 − 7𝑚 − 1) ⇒ 8𝑚2 + 12𝑚 − 5 − 2𝑚2 − (−7𝑚) − (−1) ⇒ 8𝑚2 + 12𝑚 − 5 − 2𝑚2 + 7𝑚 + 1 Step 2 ⇒ (8𝑚2 − 2𝑚2 ) + (12𝑚 + 7𝑚) + (−5 + 1) ⇒ (6𝑚2 ) + (19𝑚) + (−4) ⇒ 6𝑚2 + 19𝑚 − 4 ∴ (8𝑚2 + 12𝑚 − 5) − (2𝑚2 − 7𝑚 − 1) = 6𝑚2 + 19𝑚 − 4