Polynomial factorization #17 - Questions and Answers

Question 1 Factor completely each polynomial, and indicate any that are not factorable using integers. 1 - 16x⁴ Answer 1 Step 1:Given and formula 1 - 16x⁴ Formula a² - b² = (a + b)(a - b) Step 2:Solution 1 - 16x⁴ = (1²)² - ((4x²)²)² = (1 - (4x²)²)(1 + (4x²)²) = (1 - 4x²)(1 + 4x²)(1 + 16x⁴) = (1 + 2x)(1 - 2x)(1 + 4x²)(1 + 16x⁴) Answer: (1 + 2x)(1 – 2x)(1 + 4x²) Answer 2 Consider the polynomial 1 - 16x⁴. The above polynomial can be solved as follows. 1 - 16x⁴ = (1)² - (4x²)² Match the expression with format a² – b² = (1 - 4x²)(1 + 4x²) As a² - b² = (a + b)(a - b) = ((1)² - (2x)²) (1 + 4x²) = (1 + 2x)(1-2x)(1 + 4x²) As a² - b² = (a + b)(a - b) So the Complete factor is (1+2x) (1 – 2x)(1+4x²). Question 2 Factor completely each polynomial, and indicate any that are not factorable using integers. x³ - 9x Answer 1 Step 1 A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Factoring polynomials involves breaking up a polynomial into simpler terms (the factors) such that when the terms are multiplied together they equal the original polynomial. Step 2 The given polynomial is x³ - 9x. Simplify the polynomial and write it in factored form by taking the common terms out as follows; x³ - 9x = x(x² - 9) ... Take the common term out. = x(x² - 3²) = x(x - 3)(x + 3)... Use difference two square a²-b² = (a-b)(a+b). Hence, the given polynomial is factored completely and is equal to x³ - 9x = x(x - 3)(x + 3). Answer 2 Consider the polynomial x³ - 9x. Taking out the common factor x from the above polynomial. x³ - 9x = x(x² - 9) Now the above polynomial can be factorized as follows. x(x² - 9) = x(x² - 3²) Match the polynomial with a² – b² form = x(x + 3)(x - 3) as a² - b²= (a + b)(a - b) So the Complete factor is x(x+3)(x-3). Question 3 What is the gratest common factor of 72 and 90? Answer 1 Given numbers are 72 and 90 Prime factors of 72 = 2×2×2×3×3 Prime factors of 90 = 2×3×3×5 By definition of greatest common factor of given whole numbers, it contains each Answer 2 GCF of 72 and 90 is the largest possible number that divides 72 and 90 exactly without any remainder. The factors of 72 and 90 are 1,2,3,4,6,8,9,12,18,24,36,72 and 1,2,3,5,6,9,10,15,18,30,45,90 respectively. There are 3 commonly used methods to find the GCF of 72 and 90-long division. Euclidean algorithm, and 'factorization. Question 4 Solve quadratic equation by factoring. x² - 6x + 8 = 0 Answer 1 Step 1 Given the quadratic equation: x² - 6x + 8 = 0 Step 2 Solving by factorization: x² - 6x + 8 = 0 x² - 4x - 2x + 8 = 0 (x² - 4x) - (2x - 8) = 0 x(x-4) - 2(x-4) = 0 (x-4)(x-2) = 0 (x-4)=0 or (x-2)=0 x=4 or x=2 Hence, the solution is x=2,4 Answer 2 In the first place, we solve the equation x² + 6x + 8 = 0 by using the Quadratic Formula. Recall the formula, The Quadratic Formula, The solutions of ax² + bx + c = 0, a ≠ 0, are given by the Quadratic Formula, x = -b±√b²-4ac / 2a Set a=1, b=6 and c=8. We have x = -b ± √(b² - 4ac) / 2a x = -6 ± √(6² - 4(1)(8)) / 2(1) x = -6 ± √4 / 2 Hence we have x = (-6 ± √4) / 2 x = (-6 ± 2) / 2 Or follows that x = (-6 + 2) / 2 = -4/2 = -2 ,or x = (-6 - 2) / 2 = -8/2 = -4 In the second place, we solve x² + 6x + 8 = 0 by factoring. Factoring x² + 6x + 8 = 0, we have x² + 6x + 8 = x² + 2x + 4x + 8 = x(x+2) + 4(x+2) = (x+4)(x+2) = 0 Recall the property, Zero-Factor Property, if ab=0, then a=0 or b=0. This property also applies to three or more factors. Since (x+4)(x+2)=0, we have x+4=0 or x+2=0. Note that x+4=0 implies x=-4. Also, note that x+2=0 implies x=-2. In conclusion, we obtain the solution of the equation x² + 6x + 8 = 0 is x such that x=-4 or x=-2 Answer 3 x² - 6x + 8 = x² - 2x - 4x + 8 ⇒ x² - 6x + 8 = x(x - 2) - 4(x - 2) ⇒ x² - 6x + 8 = (x - 2)(x - 4) Thus, the given quadratic equation x² - 6x + 8 = 0 can be factorized as equation (x-2)(x-4)=0 which implies either x=2 or x=4. For x=2, we get, x² - 6x + 8 = 2² - 62 + 8 = 4 - 12 + 8 = 0 For x=4, we get, x² - 6x + 8 = 4² - 64 + 8 = 16 - 24 + 8 = 0 Result: (x-2)(x-4)=0 implies x=2 or x=4 Question 5 Use the grouping method to factor the polynomial in the box completely. Use the caret symbol (^) to write exponets; for example enter x² as x^2. Write each factor as a polynomial in descending order. 6x⁵ + 36x² + 24x Answer Step 1 We have to factor this Step 2 6x⁵ + 36x² + 24x = 6x(x⁴ + 6x + 4) hence 6x⁵ + 36x² + 24x = 6x(x⁴ + 6x + 4) Also, 6x(x⁴ + 6x + 4) = 6x¹(x⁴ + 6x + 4)