Factor the polynomial f(x) = x^4 - 3x^3 - 7x^2 + 27x - 18 given 3 and -3 are zeros

Answer:     (x+3)(x-1)(x-2)(x-3)scroll the attachment images if you're on mobile as there are  three images that show my entire answer with the proper formatting.======    f(x) = x⁴ - 3x³ - 7x² + 27x - 18if we're given that 3 and -3 are zeros, then, by the factor theorem, these two binomials must be factors of this polynomial:     x - (3), and    x - (-3), or x + 3we can first divide f(x) by x + 3. set up synthetic division. the top row has all the coefficients of the dividend. the value, a, of the binomial divisor in the form x-a is placed on the top left. here, the divisor is x + 3 = x - (-3), so we put -3 in the top left.    in the first column, bring down the 1. then multiply that 1 with the number in the top left to get -3. after that, move that product of -3 under the next column.    add that column up. since (-3) + (-3) = -6, we put -6 under that column.repeat this process until the end.    according to synthetic division, x⁴ - 3x³ - 7x² + 27x - 18  divided by x+3 has, reading the right to left, a quotient polynomial with    remainder: 0    constant: -6    coefficient on x: 11    coefficient on x²: -6    coefficient on x³: 1therefore,      which rearranges to    we have factored out one factor. since we are given that 3 is a zero, we can divide  x³  - 6x² + 11x - 6 by x-3 using synthetic division    according to synthetic division, x³  - 6x² + 11x - 6  divided by x-3 has, reading the right to left, a quotient polynomial with    remainder: 0    constant: 2    coefficient on x: -3    coefficient on x²: 1therefore, x³  - 6x² + 11x - 6 turns into (x² - 3x + 2)(x-3)  thus    x² - 3x + 2 can be factored using quadratic methods. two numbers that multiply to get +2 (constant) and add  to get -3 (coeff. on x) are -2 and -1. sox²  -  3x  +  2 factors into (x-2) and (x-1). therefore      the complete factored form of f is      (x+3)(x-1)(x-2)(x-3)

In all honesty, there really is no “work” per say to show. mainly it’s just using the process of elimination. the question is asking which domain(s) would be the most appropriate for the function c(n). n is the number of observed vehicles so right away, 1, 2, and 3 can be eliminated.why? 1, 2 start in the negatives and therefore it is impossible to observe negative cars. so, #4 would be the best answer to the given problem.