Factor down the following trinomials.1. 3x²-7x+42. 4x²+8x+43.5x²-2x-34.6x²+x-1
Answer (1)
Answer:Certainly! Here are the factorizations of the given trinomials:1. For 3x² - 7x + 4:We look for two numbers that multiply to 3 * 4 = 12 and add up to -7. These numbers are -3 and -4.Rewrite the middle term:3x² - 3x - 4x + 4Factor by grouping:3x(x - 1) - 4(x - 1)Factor out common binomial:(3x - 4)(x - 1)2. For 4x² + 8x + 4:Notice this is a perfect square trinomial:(2x + 2)² or (2(x + 1))²Factor completely:(2x + 2)² = (2(x + 1))²Alternatively, factor as a quadratic:Find two numbers that multiply to 4 * 4 = 16 and add up to 8, which are 4 and 4.Rewrite the middle term:4x + 4xFactor out common factor from each term:4(x + 1) + 4(x + 1)Factor out the common binomial:(4 + 4)(x + 1) — but since both terms have a common factor of 4, it's better to write as:4(x + 1)²3. For (5x)² - 2x -3:Rewrite as: 5x² - 2x -3Find two numbers that multiply to (5)(-3)= -15 and add up to -2. These are -5 and +3.Rewrite the middle term:5x² -5x +3x -3Factor by grouping:5x(x -1) +3(x -1)Factor out common binomial:(5x +3)(x -1)4. For (6x)² + x -1:Rewrite as: 36x² + x -1Find two numbers that multiply to (36)(-1)= -36 and add up to 1. These are likely more complex, but we can attempt factoring by trial or quadratic formula.Using quadratic formula for roots of: 36x² + x -1 =0,the roots are given by:x = [-b ± √(b²-4ac)] / (2a)take a=36, b=1, c=-1.the discriminant D = (1)² – 4*36*(-1) =1+144=145.x = [-1 ± √145] /72.hence, it does not factor nicely over integers.Thus, the factored form over rationals is:(6x)^2 + x –1 remains unfactorable with rational coefficients unless using quadratic formula for roots.Alternatively, it can be expressed as:directly in quadratic form or via approximation of roots.Please let me know if you need further assistance.Step-by-step explanation:
