Polynomial problems

Question

Find all real zeros of the polynomials and determine the multiplicity of each. (solve by factoring)

f(x) = x³ - 6x² - 9x + 54

f(x) = x⁴ + 5x³ + 6x²

Answer 1

f(x) = x³ - 6x² - 9x + 54

1. use the rational root theorem to find one root, then polynomial long division to reduce the other factor(s) to a quadratic term and then a method for solving quadratics to find the roots of that term if they exist in the real numbers.
2. since the problem states that you should solve by factoring, I suppose that you could assume that the quadratic is factorable.

f(x) = x⁴ + 5x³ + 6x²

1. Factor out the highest power of x that all the terms have in common. Then you will have reduced the other factor to a quadratic which in turn has 2 other factors. Set each factor = to zero and solve for your roots.

Answer 2

f(x) = x³ - 6x² - 9x + 54

After determining that there is no GCF to factor, since there are 4 terms in the polynomial function, consider attempting to factor by grouping.

x²(x-6) - 9(x-6) = 0 -- factoring by grouping does in fact "work" since (x-6) if a factor of both terms

(x-6)(x² - 9) = 0

You can take it from there.